GRAPH HOMOMORPHISMS THROUGH ... - Semantic Scholar
JGT 44 (2003) 15–38
GRAPH HOMOMORPHISMS THROUGH RANDOM WALKS Amir Daneshgar Department of Mathematical Sciences Sharif University of Technology P.O. Box 11365–9415, Tehran, Iran Hossein Hajiabolhassan Institute for Studies in Theoretical Physics and Mathematics (IPM) & Department of Mathematical Sciences Shahid Beheshti University P.O. Box 19834, Tehran, Iran Abstract In this paper we introduce some general necessary conditions for the existence of graph homomorphisms, which hold in both directed and undirected cases. Our method is a combination of Diaconis and Saloff– Coste comparison technique for Markov chains and a generalization of Haemers interlacing theorem. As some applications, we obtain a necessary condition for the spanning subgraph problem, which also provides a generalization of a theorem of Mohar (1992) as a necessary condition for Hamiltonicity. In particular, in the case that the range is a Cayley graph or an edge-transitive graph, we obtain theorems with a corollary about the existence of homomorphisms to cycles. This, specially, provides a proof of the fact that the Coxeter graph is a core. Also, we obtain some information about the cores of vertex–transitive graphs.
Index Words:
1
graph colouring, graph homomorphism, graph spectra, Markov chains.
Introduction
Graph homomorphisms, as natural adjacency–preserving maps between graphs, are among the most fundamental concepts in graph theory. Considering the 1 2
Correspondence should be addressed to [email protected]. This research is partially supported by the Institute for Studies in Theoretical Physics and Mathematics (IPM).
15
importance of the concept in theoretical graph theory and its profound consequences in applications, it is a natural approach, as in many other branches of mathematics, to look for no–homomorphism theorems through the study of necessary conditions which follow from the existence of such a map (it is a well-known fact that to determine the existence of such a map is usually a hard problem [25]). These conditions, in the current literature, vary from simple criteria in terms of chromatic number or odd girth of the graphs to some interesting theorems such as Albertson–Collins–Bondy–Hell no–homomorphism lemma (see Theorem B). The main objective of this paper is to introduce general no–homomorphism theorems by means of considering random walks on the corresponding graphs. To be more specific, assume the existence of an onto homomorphism σ from G to H, and consider the spectra of the corresponding random walks, which can be compared in L2 (πG ) and L2 (πH ) by means of Courant–Fischer Min-Max principle, if stationary distributions πG and πH with nonzero entries exist, respectively. The comparison technique based on this idea was first considered by P. Diaconis and L. Saloff-Coste to compare two ergodic Markov chains [9, 10, 32]. Organization of the paper is as follows. In Section 2 we go through some basic definitions and concepts from graph theory and the theory of Markov chains. Section 2.4 contains the second basic idea of the paper which has its roots in both theories, where we consider the spectrum of the quotient chain on the inverse image of σ, denoted by Gσ . The interlacing theorem for the spectrum of the quotient walk on Gσ and that of the original random walk on G is just an L2 version of the well-known contribution of Haemers [18, 20, 21, 22]. In Section 3 we apply these tools to obtain our main no–homomorphism theorems (Theorems 3–6). In Section 4 we apply our results to two important problems. Section 4.1 contains a result that can be used to test whether G is a spanning subgraph of H (Theorem 7). Also, in this section we recall a result of Saloff–Coste which can be used as another test of this type (see Corollary 4.2.6 of [32]). Also, in Section 4.2 we apply our results to obtain some information about the cores of vertex–transitive graphs, which introduces some bounds for the order and the index of regularity of such cores (Theorem 9). Moreover, we give a proof of the fact that the Coxeter graph is a core. Section 5 contains our concluding remarks.
2 2.1
Basic definitions and concepts Some preliminaries from graph theory
In this section we recall some basic concepts and notations. Also, we refer to the standard literature for what does not appear here (e.g. [33]). 16
A directed graph G (or a digraph for short) on n vertices is a structure which is determined by an ordered pair (V (G), E(G)) such that V (G) = {v1 , ..., vn } is the set of n vertices and E(G) is a family of ordered pairs (u, v), for some elements u and v of V (G), called directed edges. In this paper we only consider finite (di)graphs. Also, we identify the concepts of a simple graph (or a graph for short) and a symmetric, loopless digraph. In what follows we use the notation e = uv to denote the edge e = (u, v), where u → v and u ↔ v are used for directed and simple connections, respectively. Also, we define, →
def
E (A, B) = {uv ∈ E(G) | u ∈ A & v ∈ B & u → v}. +
Moreover, dG (v) (res. dG (v)) denotes the degree (res. out-degree) of the vertex + + v in G, and ∆G (δG ) (res. ∆G (δG )) denotes the maximum (minimum) degree (res. out-degree) of G. Kn is the complete graph on n vertices and Cn (res. → C n ) is the (res. directed) cycle on n vertices. A homomorphism f : G −→ H from a digraph G to a digraph H is a map f : V (G) −→ V (H) such that u → v implies f (u) → f (v). Similarly, a homomorphism f : G −→ H from a graph G to a graph H is a map f : V (G) −→ V (H) such that u ↔ v implies f (u) ↔ f (v). Hom(G, H), Homv (G, H) and Home (G, H) denote the sets of ordinary, onto (vertices) and onto–edges homomorphisms from G to H, respectively. A Cayley digraph Cay(G, X) of a group G with respect to the subset X ⊆ G − {0}, in which 0 is the identity element of G, has the vertex set G, and g → h if hg −1 ∈ X. It is easy to see that Cay(G, X) is connected if and only if X is a generator of G, and also Cay(G, X) is symmetric if and only if x ∈ X implies that x−1 ∈ X. Moreover, we define Cays (G, X) to be the symmetric Cayley graph Cay(G, X ∪ X −1 ). It easily follows from the definition that every Cayley graph is vertex–transitive. If H is a subgraph of G, then a retraction of G to H is a homomorphism r : G −→ H such that r(x) = x for all x ∈ V (H). A graph is a core (or a minimal graph [16]) if it does not admit a homomorphism to a proper subgraph of itself. It is easy to see [16, 24] that every graph G contains a unique (up to isomorphism) subgraph H which is a core and admits a retraction r : G −→ H. • Then usually this subgraph H is called the core of G, and is denoted by G . Also note that in this situation for each graph K, there exists a homomorphism • f : K −→ G if and only if there exists a homomorphism f : K −→ G . Theorem A. [19, 23] Let G be a vertex–transitive graph. Then, •
• G is a vertex–transitive graph. •
•
• If σ ∈ Hom(G, G ), then for all x ∈ V (G ), the inverse images σ −1 (x) (G)| have the same cardinality, namely |V|V(G • . )| 17
A digraph (graph) G is r–critical if χ(G) = r and χ(G−e) < r for each edge e ∈ E(G) (We also assume that χ(G) for any digraph G is the chromatic number of the simple graph obtained by changing all directed edges to simple ones and excluding all loops). Also, r–vertex–critical graphs are defined similarly. In [4] Bondy and Hell define ν(G, K) for two graphs G and K as the maximum number of vertices in a subgraph of G that admits a homomorphism to K; and using this they give the following generalization of a result of Albertson and Collins [1] in which µ(G, K) = |V (G)|/ν(G, K). Theorem B. [4] Let G, H, K be graphs where H is vertex–transitive. If there exists a homomorphism f : G −→ H then µ(G, K) ≤ µ(H, K).
2.2
Random walks on graphs
In this section we go through some classical known results about random walks on digraphs (e.g see [2, 5, 32]). It should be noted that all we need throughout this paper is a random walk KG on the base graph G with a stationary distribution πG such that πG (v) 6= 0 for any vertex v ∈ V (G) (by abuse of language, we call this a nowherezero stationary distribution). However, we always confine ourselves to the natural random walk on a finite graph, as an important special case, to illustrate our results. The natural random walk on a digraph G can be defined through the kernel ( 1 u→v + d (u) KG (u, v) = G 0 u 6→ v, where one can also think of KG (u, v) as the entry of a |V (G)| × |V (G)| matrix KG which is indexed by the row index u and the column index v. Note that the matrix KG is not necessarily symmetric even if the corresponding graph is a symmetric digraph, however, for regular digraphs, and of course for regular graphs as a special case, the corresponding matrix is symmetric. On the other hand, it is a simple observation that this chain is irreducible (i.e. for any u, v there exists a power k = k(u, v) such that KGk (u, v) > 0) if and only if G is a strongly connected digraph; and it is well-known that there exists a unique nowherezero stationary distribution π for such a finite Markov chain P (i.e. πKG = π, v π(v) = 1 and for each v ∈ V (G) we have π(v) 6= 0). As a matter of fact, the methods which are used in this paper are very much related to evaluation or bounding the rate of convergence to the stationary distribution (e.g. see [5, 9, 10, 11, 12, 14, 15, 27, 32]). Definition 1. define
For a random walk KG with stationary distribution πG we def
• QG (u, v) = KG (u, v)πG (u). 18
M
def
m
def
• πG = max πG (u) and πG = min πG (u). u∈V (G)
M
def
• QG =
u∈V (G)
def
m
max u6=v∈V (G)
QG (u, v) and QG =
min {QG (u, v) | QG (u, v) 6= u6=v∈V (G)
0}. Also, KG is called reversible if ∀ u, v ∈ V (G) QG (u, v) = QG (v, u).
(1) ♠
The following lemma is stated for further reference. Lemma 1. Let G be a strongly connected digraph whose natural random walk has the stationary distribution πG . Then • If G is Eulerian, +
M
πG
∆G , = |E(G)|
+
δG πG = , |E(G)|
QG = QG =
δG , 2|E(G)|
QG = QG =
m
M
m
1 . |E(G)|
• If G is simple, M
πG =
2.3
∆G , 2|E(G)|
m
πG =
M
m
1 . 2|E(G)|
Dirichlet forms and L2 (πG )
Let KG be the kernel of a random walk on V (G) with a nowherezero stationary distribution πG . We consider KG as a linear operator on L2 (πG ) of all real functions on V (G) with the following inner product def
< f, g >πG =
X
f (v)g(v)πG (v).
v∈V (G)
Consequently 12
kf kπG =
X
|f (v)|2 πG (v) ,
v∈V (G)
and it is easy to see that the adjoint of KG is defined by the kernel KG∗ (u, v) =
19
QG (v, u) , πG (u)
where it is also clear that KG is the kernel of a reversible Markov chain if and only if it is selfadjoint in L2 (πG ) i.e. KG = KG∗ . Let SG = I − 21 (KG + KG∗ ) be the additive symmetric part of KG where I is the identity operator. Then, EG (f, f )
def
= < SG (f ), f >πG =< (I − KG )f, f >πG =
X
1 2
(2)
|f (u) − f (v)|2 QG (u, v),
u,v∈V (G)
is called the Dirichlet form of KG . Moreover, all eigenvalues of SG (or those of the Dirichlet form of KG by abuse of language) are real and positive. Hence, if |V (G)| = n, then we order the eigenvalues of SG as follows G
G
G
G
0 = λ0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λn−1 . On the other hand, it is well-known that if the base graph is strongly connected (i.e. the Markov chain is irreducible) then by Perron–Frobenius theorem and G G the fact that SG is selfadjoint one may deduce 0 = λ0 < λ1 . Also, by Courant– Fischer Min-Max principle (see [26, 32]) for any 0 ≤ k < n one may write ) ( ) ( EG (f, f ) EG (f, f ) G λk = min max = max min , kf k2π kf k2π W ∈W 06=f ∈W W ∈W ⊥ 06=f ∈W k+1 k
G
G
in which def
Wk = {W ≤ L2 (πG ) | dim(W ) ≥ k},
def
Wk⊥ = {W ≤ L2 (πG ) | dim(W ⊥ ) ≤ k}.
It should be noted that in the classical theory of ergodic Markov chains the eigenvalues of the Dirichlet form are used to estimate the rate of convergence G of the chain. Note that if KG is selfadjoint then λ1 is the second smallest eigenvalue of SG = I − KG . The following theorem was used by Diaconis and Saloff-Coste in their comparison technique [9, 10, 32]. Theorem 1. [32] Let G and H be two digraphs with |V (G)| = n and |V (H)| = m ≤ n whose random walks have nowherezero stationary distributions πG and G H πH , respectively. Also, let λk and λk be the corresponding eigenvalues of SG and SH , respectively, and let T : L2 (πH ) −→ L2 (πG ) be a linear map. Then, if for some constants a > 0 and A < ∞ and for any f ∈ L2 (πH ) we have akf k2π ≤ kT (f )k2π H
then
a G λ A k
G
and EG (T (f ), T (f )) ≤ A EH (f, f ),
H
≤ λk for any 1 ≤ k ≤ m − 1.
Proof. Note that T is one to one and apply Courant–Fischer Min-Max principle (see [26, 32]). 20
It is also instructive to note that if one defines X X def def πG (f ) = πG (v)f (v) and VarπG (f ) = |f (v) − πG (f )|2 πG (v), v∈V (G)
v∈V (G)
then one may also write G
λ1
= min
EG (f, f ) | VarπG (f ) 6= 0 VarπG (f )
n o = min EG (f, f ) | kf kπG = 1, πG (f ) = 0 .
(3)
Of course, it is usually hard to compute the whole spectrum when |V (G)| is large, and one should usually obtain bounds or estimates for the eigenvalues G λk . This is a basic subject in the theory of Markov chains.
2.4
The quotient walk
The content of this section is actually a culmination of ideas from algebraic combinatorics and probability theory. Assuming a graph homomorphism σ ∈ Homv (G, H), our basic objective is to reduce the size of the state space of a random walk KG on the base graph G to a new (quotient) random walk on the base graph Gσ , in such a way that |V (Gσ )| = |V (H)|. To do this, we use a more restricted version of the technique used in Theorem 1 by choosing the linear map to be a unitary operator. This has already been done for symmetric matrices in the celebrated work of Haemers [18, 20, 21, 22]. Let σ ∈ Homv (G, H) where G and H are digraphs such that n = |V (G)| ≥ |V (H)| = m and assume that the corresponding random walks have nowherezero stationary distributions πG and πH , respectively. For any x ∈ V (H) define, def
def
Ux = σ −1 (x) and U = {Ux | x ∈ V (H)}, and note that each Ux is an independent set of vertices. Then we define the (nowherezero) probability distribution πU on the set U as X πG (v). πU (Ux ) = v∈Ux
Define the operator P : L2 (πU ) −→ L2 (πG ) through the following kernel 1 v ∈ Ux P (v, Ux ) = 0 otherwise,
(4)
and note that if we think of P as an n × m matrix, then the column which is indexed by Ux is actually the characteristic function of this set on V (G). 21
Now, it is easy to see that πU = πG P , and if we define P ∗ : L2 (πG ) −→ L2 (πU ) through the kernel def π (v) P ∗ (Ux , v) = G P (v, Ux ), πU (Ux ) then P ∗ is actually the adjoint of P i.e., < f, P (g) >πG =< P ∗ (f ), g >πU . Also, P ∗ P = IπU which shows that P is a unitary operator i.e., kP (f )kπG = kf kπU for any f ∈ L2 (πU ). σ On the other hand, consider the quotient random walk on G defined on σ def V (G ) = U through the kernel P ∗ KG P (we call this the quotient chain on the inverse image of σ). Then we have SGσ = SP ∗ K
G
P
1 = I − (P ∗ KG P + P ∗ KG∗ P ) = P ∗ SG P, 2
(5)
σ
and also the kernel of the quotient walk on G can be described explicitly through the following equation, X X QGσ (Ux , Uy ) = QG (u, v), u∈Ux
v∈Uy σ
which shows that the transition probabilities of the quotient walk on G from Ux to Uy is actually a weighted mean of the transition probabilities (e.g. number of edges when G is regular) in the original walk on G from Ux to Uy . Theorem 2. Let σ ∈ Homv (G, H) where G and H are digraphs with n = |V (G)| ≥ |V (H)| = m and nowherezero stationary distributions πG and πH , respectively. Then, ∀ 1≤k ≤m−1
G
σ G
G
λk ≤ λk ≤ λn−m+k .
Proof. This is again a direct application of Courant–Fischer Min-Max principle (see [18, 20, 21, 22, 26]) to compare the eigenvalues of SG and SGσ = P ∗ SG P (see Equation 5). Just note that in this case P is one to one since it is unitary. Moreover, note that the right hand inequality follows from the left hand one by considering the operator −SG .
3
No–homomorphism theorems
Let σ ∈ Homv (G, H) and assume that the corresponding random walks have nowherezero stationary distributions πG and πH , respectively. What we are going to show in this section is that there are relationships between the spectra 22
of SG and SH . Clearly, this will provide no–homomorphism theorems which can be used later in our special applications. σ The main setup is to consider the quotient random walk on G and then compare it with the random walk of H. This will give rise to necessary conditions in terms of the corresponding spectra (specially the spectral gap). Using these ideas we first focus on general onto homomorphisms, then we generalize our results to arbitrary homomorphisms and we also consider some special cases under some regularity or symmetry conditions.
3.1
Spectral conditions for onto homomorphisms
The comparison methods we use in this section are mainly developed by P. Diaconis and L. Saloff-Coste, however, we consider the whole spectrum to get more information [9, 10, 14, 32]. We begin by a couple of definitions. Definition 2. If σ ∈ Homv (G, H) we define, def
• Mσ =
min
→
→
{|E (σ −1 (x), σ −1 (y))| | E (σ −1 (x), σ −1 (y)) 6= ∅}.
x6=y∈V (H) σ
→
def
max |E (σ −1 (x), σ −1 (y))|.
• M =
x6=y∈V (H)
def
σ
def
• Sσ = min |σ −1 (x)| and S = max |σ −1 (x)|. x∈V (H)
x∈V (H) σ
def M Sσ
def
σ and θ = M . Sσ inf{ησ | σ ∈ Homv (G, H)} v def • η (G, H) = −∞ σ sup{θ | σ ∈ Homv (G, H)} v def • θ (G, H) = +∞
• ησ =
e
σ
Homv (G, H) 6= ∅ Homv (G, H) = ∅ Homv (G, H) 6= ∅ Homv (G, H) = ∅
e
Also, η (G, H) and θ (G, H) are define in terms of Home (G, H), similarly. ♠ The following proposition which is mainly based on Theorem 1 provides the main comparison lemma in our further applications. Proposition 1. Let σ ∈ Homv (G, H) where G and H are digraphs with n = |V (G)| ≥ |V (H)| = m and nowherezero stationary distributions πG and πH , respectively. Then for 1 ≤ k < m σ G
λk
M
M
Q π H ≤ ησ Gm Hm λk . QH πG
Also, if σ ∈ Home (G, H), then for 1 ≤ k < m σ G
λk
m
m
QG πH H ≥ θ M λk . QM π H G σ
23
Proof. First assume that σ ∈ Homv (G, H). Note that since σ is a graph homomorphism, for all x 6= y QGσ (Ux , Uy ) 6= 0 ⇒ QH (x, y) 6= 0. Also, from Equation 2 we have EGσ (f, f ) =
1 2
X
|f (y) − f (x)|2 QGσ (Ux , Uy )
x,y∈V (H)
and EH (f, f ) =
1 2
X
|f (y) − f (x)|2 QH (x, y).
x,y∈V (H)
But, since σ
M
QGσ (Ux , Uy ) ≤ M QG , we may write M
QG EGσ (f, f ) ≤ M EH (f, f ). Qm H σ
On the other hand, similarly, we have m
2
kf kπ
σ G
πG ≥ Sσ M kf k2π . H πH
Consequently, by considering the identity map ι : U −→ V (H) where U is the σ def vertex set of G , T (f ) = f ◦ ι and Theorem 1 for 1 ≤ k < m we have, σ G
λk
M
M
Q π H ≤ ησ Gm Hm λk . QH πG
Conversely, if σ ∈ Home (G, H), also for all x 6= y we have, QH (x, y) 6= 0 ⇒ QGσ (Ux , Uy ) 6= 0. Hence, using the same method we get σ G
λk
m
m
QG πH H ≥ θ λ . QM πGM k H σ
Now, we are ready to obtain one of our main no–homomorphism theorems. Theorem 3. Let G and H be two digraphs with n = |V (G)| ≥ |V (H)| = m such that πG and πH are nowherezero stationary distributions, respectively. Then 24
a ) If Homv (G, H) 6= ∅, then for all 1 ≤ k ≤ m − 1, M
M
Q π H λk ≤ η (G, H) Gm Hm λk . QH πG G
v
b ) If Home (G, H) 6= ∅, then for all 1 ≤ k ≤ m − 1, m
G
λn−m+k Proof.
m
Q π H ≥ θ (G, H) MG HM λk . QH πG e
Apply Proposition 1 and Theorem 2.
It should be noted that the full power of Theorem 2 appears in the second part, since the first part may also be obtained by a direct comparison of KG and KH through the pullback along σ. The following corollary covers many important cases since any simple graph is an Eulerian digraph as a symmetric loopless digraph. Corollary 1. Let G and H be strongly connected Eulerian digraphs with n = |V (G)| ≥ |V (H)| = m. Then a ) If Homv (G, H) 6= ∅, then for all 1 ≤ k ≤ m − 1, +
G
v
λk ≤ η (G, H)
∆H +
δG
H
λk .
b ) If Home (G, H) 6= ∅, then for all 1 ≤ k ≤ m − 1, +
G
e
λn−m+k ≥ θ (G, H) Proof.
δH
+
∆G
H
λk .
Apply Lemma 1.
Considering the case of regular graphs is instructive. Note that in this case G for an s–regular graph sλk is the kth eigenvalue of the combinatorial Laplacian. Therefore, from this point of view SG can be thought of as a generalized (normalized) version of the Laplacian of G.
3.2
Estimations and some finer tools
In this section we focus on some basic methods that can be applied to obtain v e information about the parameters η (G, H) and θ (G, H) which are needed in Theorem 3. Also, we consider methods that can be used to get rid of some of these parameters under special conditions. Needless to say, methods 25
considered in this section are just some examples of a variety of approaches that can be chosen to apply Theorem 3. Note that as first estimates, we have the following rough bounds, v
e
η (G, H) ≤ β2 (G) and α(G)−1 ≤ θ (G, H), where α(G) is the independence number of G and β2 (G) is the largest integer ε, such that there exists a subgraph G0 of G with |E(G0 )| = ε and Homv (G0 , K2 ) 6= ∅. Of course, there are a lot of known results and methods to obtain estimates for α(G) and β2 (G) (specially for simple graphs), however, getting better bounds asks for some more information about G. As an important example we mention the following simple lemma, which also shows the vast range of the cases Theorem 3 can be applied. Lemma 2. Let G and H be two digraphs, then a ) If H is vertex–critical with χ(H) = χ(G), then Hom(G, H) = Homv (G, H) and Sσ is greater than or equal to the minimum number of vertices which is needed to reduce the chromatic number of G. b ) If H is edge–critical with χ(H) = χ(G), then Hom(G, H) = Home (G, H) and Mσ is greater than or equal to the minimum number of edges which is needed to reduce the chromatic number of G. In the next example we consider the set Hom(P10 , C5 ), of all homomorphisms from the Petersen graph P10 to the 5–cycle C5 . Although, it will easily follow from our next results that Hom(P10 , C5 ) = ∅, but we follow the details of the next example to show how one may estimate the necessary parameters. Example 1. Let us consider a homomorphism σ ∈ Hom(P10 , C5 ). First, note that since P10 contains two disjoint copies of C5 , we can deduce that σ S = 2 (note that this also follows from the fact that P10 is vertex–transitive and that C5 is a core). Since C5 is critical, the homomorphism should be in Home (P10 , C5 ), and by Lemma 2(b) we have Mσ ≥ 3. Hence, applying Theorem 3(b) for k = 3 shows that we should have 1.5 ≤
Mσ e σ ≤ θ (G, H) ≤ 1.38, S
which is a contradiction. Hence, Hom(P10 , C5 ) = Home (P10 , C5 ) = ∅. (Note that Theorem 3(a) is not applicable in this case.) ♣ Example 2. In this example we consider the digraphs G = Cay(Z22 , {2, 3, 4, 5, 6, 7, 8, 9}) and H = Cay(Z11 , {1, 2, 3}), 26
where we consider the existence of a homomorphism σ in Hom(G, H). First, we note it is not hard to check that χ(G) = χ(H) = 6, and that H is a 6–critical digraph; which shows that Hom(G, H) = Home (G, H). Also, it can be checked that we need to delete at least two vertices of G to reduce its chromatic number to 5. Hence, by Lemma 2 we have Sσ ≥ 2, and consequently, since any homomorphism should be onto the vertices of H we σ σ have Sσ = S = 2. Moreover, trivially, Mσ ≥ 1 and M ≤ 4. Now, applying Theorem 3(a) for k = 1 we have, σ
M 3.76 ≤ η (G, H) ≤ ≤ 2, Sσ v
which is a contradiction. Therefore, Hom(G, H) = Home (G, H) = ∅. First, note that Theorem 3(b) is not applicable in this case. Also, note that since both G and H are regular digraphs, the above inequality also holds for the ˜ = Cays (Z22 , {2, 3, 4, 5, 6, 7, 8, 9}) and H ˜ = Cays (Z11 , {1, 2, 3}). simple graphs G e ˜ ˜ ˜ ˜ Hence, we have Hom(G, H) = Hom (G, H) = ∅. ♣ As it is clear, one of the main problems in applying Theorem 3 is to obtain v e good estimates for parameters η and θ . The following theorem presents one σ of the techniques that may be used to simplify this by excluding Sσ and S from computations. Theorem 4. Let G and H be two digraphs with n = |V (G)| ≥ |V (H)| = m, where πG and πH are nowherezero stationary distributions, and H = Cay(V (H), X) is a Cayley graph in which X is closed under conjugation. Then if σ ∈ Home (G, H), for all 1 ≤ k ≤ m − 1 we have, m m QGH πH H mMσ GH λk , λm(n−1)+k ≥ 1 + M n QM πGH H where GH is the cartesian product of the graphs G and H. Proof. Let V (G) = {v1 , . . . , vn }, V (H) = {x1 , . . . , xm }, and consider a homomorphism σ ∈ Home (G, H). We define a map σ ˜ : V (GH) −→ V (H) as follows, σ ˜ ((vi , xj )) = σ(vi )xj i = 1, . . . , n and j = 1, . . . , m. First, we show that σ ˜ is a homomorphism. Let xj → xk in H. Consequently, xk x−1 ∈ X and (v , xj ) → (vi , xk ) in GH. But, since X is closed under i j conjugation we have, (σ(vi )xk )(σ(vi )xj )−1 = σ(vi )(xk x−1 )σ(vi )−1 ∈ X, j which shows that σ(vi )xj → σ(vi )xk in H. On the other hand, let vj → vk in G and consequently, (vj , xi ) → (vk , xi ) in GH. Again, since σ is a homomorphism we have, (σ(vk )xi )(σ(vj )xi )−1 = σ(vk )σ(vj )−1 ∈ X, 27
which shows that σ(vj )xi → σ(vk )xi in H. These show that σ ˜ is a homomorphism of graphs. σ ˜ Now, we show that S = Sσ˜ = n. To see this, consider a fixed vertex xt ∈ V (H). Then, if (vi , xj ) ∈ σ ˜ −1 (xt ), we may compute xj as xj = σ(vi )−1 xt , since V (H) is a group. Hence, |˜ σ −1 (xt )| ≥ n. However, since σ ˜ ∈ Homv (GH, H) and |GH| = mn we have |˜ σ −1 (xt )| = n for any t = 1, . . . , m. This shows σ ˜ that S = Sσ˜ = n. To estimate Mσ˜ fix an edge xt → xs in H. Then, for any 1 ≤ j ≤ m we have, σ ˜ −1 (xj ) = {(vi , σ(vi )−1 xj ) | vi ∈ V (G)}. We consider two cases, • The first components are equal. In this case vi is fixed. Also, since X is closed under conjugation we have σ(vi )−1 xt → σ(vi )−1 xs which shows that (vi , σ(vi )−1 xt ) → (vi , σ(vi )−1 xs ) in GH. This shows that we have n edges between σ ˜ −1 (xt ) and σ ˜ −1 (xs ) of this type. • The second components are equal. In this case we want to estimate the number of edges of the type (vi , σ(vi )−1 xt ) → (vj , σ(vj )−1 xs ) where σ(vi )−1 xt = σ(vj )−1 xs
(i 6= j).
This means that we should have σ(vj )σ(vi )−1 = xs x−1 ∈X t
(i 6= j) (∗).
Now if σ ∈ Home (G, H), then there are vertices vi and vj such that vi → vj in G, σ(vi ) = xt and σ(vj ) = xs ; which means that (∗) has at least one solution. Consequently, there are at least Mσ solutions for (∗) in this way which come from edges in the inverse image of σ. On the other hand, for each fixed x ∈ V (H) we have xt x → xs x in H; and again by considering the inverse image of σ we find another bunch of Mσ solutions for (∗). This shows that there are at least mMσ edges between σ ˜ −1 (xt ) and σ ˜ −1 (xs ) of this type when σ ∈ Home (G, H). These show that Mσ˜ ≥ n+mMσ and the rest of proof follows from Theorem 3. First, note that for the cartesian product as well as many other different products, the eigenvalues can be computed explicitly in terms of the eigenvalues of the factors (for the definition of a NEPS of graphs and its eigenvalues see [7, 8]). Moreover, the proof of Theorem 4 can be sharpened when H is a (directed) cycle as follows, in which there is no need to have an estimate for Mσ . 28
Corollary 2. Let G be a digraph such that such that πG is a nowherezero → stationary distributions, n = |V (G)| and ε = |E(G)|. Also, let H ∈ {Cm , C m } with m ≤ n. Then if σ ∈ Home (G, H), for all 1 ≤ k ≤ m − 1 we have, m
GH
λm(n−1)+k ≥ Proof. have
m
ε QGH πH H 1+ λk . M n QM πGH H
Note that in cycles as Cayley graphs for any fixed edge x → y we {xz → yz | z ∈ V } = E.
So far we have focused on onto homomorphisms. Now, we introduce another variant which gives rise to a necessary condition for the existence of a (not necessarily onto) homomorphism that can be used to prove many different and general no-homomorphism theorems. In what follows we propose two such applications. Theorem 5. Let G and H be two digraphs such that |V (G)| = n, |V (H)| = m and πG is a nowherezero stationary distribution. Also, assume that the group of automorphisms of H, Aut(H), acts transitively on both V (H) and E(H). If σ ∈ Hom(G, H) then, m
G
λn−1
2|E(G)|QG H ≥ λm−1 . nπGM G
H
Specially, if G is s-regular then λn−1 ≥ λm−1 . Proof.
Let |Aut(H)| = t, Aut(H) = {ζi | i = 1, . . . , t} and define, t
def
˜ = G
[
Gi ,
i=1
˜ such as G , is an isomorphic copy of where each connected component of G, i G. Also, define the homomorphism σ ˜ such that its restriction to Gi is ζi ◦ σ. σ |V (G)| |E(G)| × t and S = |V × t. It is easy to see that σ ˜ ∈ Home (G, H), Mσ = |E(H)| (H)| Moreover, if vi ∈ V (Gi ) is the vertex corresponding to the vertex v ∈ V (G), def π (v) then π ˜G˜ (vi ) = Gt defines a nowherezero stationary distribution π ˜G˜ for the ˜ natural random walk of G. The rest of the proof follows from Theorem 3(b).
29
It is easy to see that this theorem proves the following well-known lower bound for the chromatic number of an s-regular graph (in which µn is the least eigenvalue of the adjacency matrix of G), χ(G) ≥ 1 −
s . µn
Also, note that using the same method one may prove other different variants of this theorem. In what follows we mention one of these cases for its possible applications in graph decompositions. Theorem 6. Let G and H be two digraphs such that |V (G)| = n, |V (H)| = m and πG is a nowherezero stationary distribution. Also, assume that H is vertextransitive and r-regular. If σ ∈ Home (G, H) then, m
G
λn−1
rmMσ QG H ≥ λm−1 . nπGM
First, note that the number of eigenvalues that can be compared in theorems ˜ like this, heavily depends on the number of copies used to form the graph G. ˜ to H with using just Hence, when one can construct homomorphisms from G a small number of copies of G, then one may be able to use more inequalities. Also, note that the same method can be used in the case of onto (vertices) homomorphisms if we know the number of copies explicitly. We would like to emphasize that this type of general homomorphism theorems can be quite useful in the study of graph decompositions and the study of other concepts in graph theory which are defined in terms of graph homomorphisms. In the next section we just mention a simple case which is related to the spanning subgraph problem.
4 4.1
Some applications Spanning subgraphs and Hamiltonian cycles
An important case is when G is a spanning subgraph of H. Note that in this σ case considering σ as the identity homomorphism we have Sσ = M = 1. Hence, we have the following as a corollary of Theorem 3. Theorem 7. Let G and H be two digraphs with |V (G)| = |V (H)| = n such that πG and πH are nowherezero stationary distributions, respectively. If there exists 1 ≤ k < n such that M
M
Q π H λk > Gm Hm λk , QH πG G
then G is not a spanning subgraph of H. 30
Example 3. To show that the Petersen graph is not Hamiltonian, we should P show that Homv (C10 , P10 ) = ∅, which follows from Theorem 7 for λ5 10 . ♣ Note that this theorem generalizes a theorem of Mohar [30, 31]. Example 4. In this example we consider the existence of homomorphisms to circular graphs Gnd , which are defined on the vertex set V (Gnd ) = {0, . . . , n − 1} in which i ↔ j if d ≤ |i − j| ≤ n − d (for applications and more results see [34]). As an special case we note that the Kneser graph K(2, 6), is not a spanning K(2,6) subgraph of G15 , by applying Theorem 7 for λ10 . ♣ 4 Getting a similar result by comparison from the other way round is not easy in general, however, when H and G are regular graphs we have the following proposition which is essentially the Corollary 4.2.6 of [32]. We add the proof for completeness and to emphasize the application of flows to the subject. Theorem 8. Let G and H be two regular graphs with |V (G)| = |V (H)| = n, where G is edge-transitive and πG and πH are nowherezero stationary distributions, respectively. If G is a spanning subgraph of H, then |E(H)| G H λk ≥ X λk , dG (x, y)2
1 ≤ k ≤ n − 1.
x,y∈V (G)
Proof. Let ΓG (ΓG (x, y)) be the set of all paths in G (from x to y), and also let GG (x, y) ⊆ ΓG (x, y) be the set of all geodesics (shortest paths) from x to y. We use a comparison through the identity map from L2 (πG ) to L2 (πH ). But, first we note that • For each xy ∈ E(H) and each γ ∈ GG (x, y) by Cauchy–Schwarz inequality we have X |f (y) − f (x)|2 ≤ |γ| |f (v) − f (u)|2 . uv∈γ
• If we define Φ on ΓG as ( def
Φ(γ) =
QH (x,y) #GG (x,y)
0
γ ∈ GG (x, y) otherwise,
Then it is clear that (i.e. Φ is a flow), X ∀ xy ∈ E(H), x 6= y, γ∈Γ (x,y) G
31
Φ(γ) = QH (x, y).
Therefore, X
|f (y) − f (x)|2 QH (x, y) ≤
|γ|
X
|f (v) − f (u)|2 Φ(γ).
uv∈γ
γ∈Γ (x,y) G
Hence, EH (f, f ) =
X
1 2
|f (y) − f (x)|2 QH (x, y)
x,y∈V (H)
≤
1 2
X
X
|γ|
X
|f (v) − f (u)|2 Φ(γ)
uv∈γ
x,y∈V (H) γ∈Γ (x,y) G
def |γ|Φ(γ) EG (f, f ) = M EG (f, f ),
X
≤ (QG (x0 , y0 ))−1
x0 y0 ∈γ∈Γ G
for some fixed edge x0 y0 ∈ E(G). On the other hand, X M = (QG (x0 , y0 ))−1 |γ|Φ(γ) x0 y0 ∈γ∈Γ G
=2
X
X
e∈E(G)
≤2
X
|γ|Φ(γ)
e∈γ∈Γ G
X
X
e∈E(G) x,y∈V (G)
≤
1 |E(H)|
X
e∈γ∈G (x,y) G
dG (x, y)QH (x, y) #GG (x, y)
dG (x, y)2 .
x,y∈V (G)
Also, for the corresponding norms we have, X 1 kf kπH = |f (x)|2 = kf kπG . |V (G)| x∈V (G)
Consequently, by Theorem 1 we have X 1 H G λk ≤ dG (x, y)2 λk |E(H)|
∀ k ∈ {1, . . . , n − 1},
x,y∈V (G)
which completes the proof.
4.2
On vertex transitive cores
In what follows we are going to apply our results of Section 3 to obtain some information about cores. Let G be a vertex–transitive graph on n vertices, and let def O(n) = {m ∈ N | m is odd and divides n}. 32
First, we note that by Theorem A, if G is 3–regular, and there is no homomorphism from G to the cycle Cm for all m ∈ O(n), then G is a core. This can be effectively verified using Theorems 1 and 4. To show this, first we may note that Example 1 can be considered as an application of Corollary 1 to prove that the Petersen graph is a core. This also σ could be verified by Corollary 2 without estimating the parameters S and Mσ . We consider the same idea in the following example which is about the Coxeter graph. Example 5. By applying Corollary 2 to Hom(G, C7 ), where G is the Coxeter graph and k = 6, we get 2 3 1.843 ≥ (1 + ) × × 1.9 = 1.9, 2 5 which shows that Hom(G, C7 ) = ∅. This proves that the Coxeter graph is a core. It is also interesting to note that the only other proof of this, that the authors are aware of, uses the fact that any connected 2–arc transitive non–bipartite graph is a core [19, 23]. ♣ Theorem 9. Let G be a simple s–regular vertex–transitive graph with |V (G)| = n = rm whose core is a t–regular subgraph, through the homomorphism σ, with • r σ |V (G )| = m. Also, let M ≥ M and Mr ≤ Mσ . Then, t G r G ∀1≤k<m rλk ≤ M (λn−m+k + − 1), s and t G G ∀1≤k<m rλn−m+k ≥ Mr (λk + − 1). s Proof. First, note that if H is an induced t–regular subgraph of an s–regular graph G, then by considering the adjacency matrices and the interlacing lemma we have, ∀ 1≤k ≤m−1
G
H
G
s(1 − λn−m+k ) ≤ t(1 − λk ) ≤ s(1 − λk ),
in which |V (H)| = m ≤ |V (G)| = n. • Now, let |V (G )| = m and |V (G)| = rm for some integer r ≥ 1. Hence, • σ considering the corresponding homomorphism σ : G −→ G we have M ≤ r M , Sσ = r; and, consequently, by Corollary 1, r
r
tM G• tM s G λk ≤ (1 − (1 − λn−m+k )); sr sr t which proves the first inequality. The second inequality is also proved similarly. ∀ 1≤k ≤m−1
G
λk ≤
Note that the theorem essentially shows that m, the order of the core, can not be arbitrary small. Also, it is clear that the theorem provides upper and lower bounds for t, the degree of regularity of the core. 33
5
Concluding remarks
It is instructive to add some notes on the whole setup we have introduced so far and point out some important aspects that can be considered in forthcoming research. First, we wish to emphasize that the method introduced in this paper works when • There are two (not necessarily natural) random walks KG and KH such that their base graphs are G and H respectively, and there are (not necessarily unique) stationary distributions πG and πG which are nonzero on each vertex. • The operator SG is a well-behaved selfadjoint operator obtained from KG ∗ def (e.g. one may consider many other variations such as SG = I − KG KG ). • One may use different perturbation methods, such as adding loops, to define various random walks and use Theorem 3 more effectively (e.g. note that by adding loops properly, one may change a nonregular graph to a regular one). It is evident from our approach that any kind of information about the spectrum of the natural random walk of a graph has important consequences on what we may deduce about its homomorphic images or the graphs that have homomorphisms to it. This, in a special case, may draw ones attention to expanders and Ramanujan graphs as an important class of random–like graphs with a relatively large spectral gap. Even it is quite interesting to consider families of s–regular√graphs for which the spectral gap (in the classical sense) is greater than s−2 s − 1+ for some > 0 (note that by Alon–Boppana theorem there are only a finite number of such graphs for a fixed s (see [3, 6, 28, 29] and references therein)). Also, one should note that, since it is not always easy to compute the whole spectrum, it is usual to focus on the spectral gap, λ1 , or some other parameters which are mainly related to the rate of convergence of the chain, like the log–Sobolev or the Cheeger constants. It is interesting to note that one may also use the techniques of the previous sections to obtains some partial results concerning these parameters too. We also note that the role of these parameters can be quite different from the spectral gap, which, in a way, justifies their consideration as a separate subject (for more on these subjects see [11, 12, 15, 17, 32]). For instance, we may consider one special type of the Cheeger constant as follows. Definition 3. Let G be a digraph whose natural random walk has the nowherezero stationary distribution πG . Then 34
• For any edge e = uv we define 1 def (QG (u, v) + QG (v, u)) and df (uv) = f (v) − f (u). 2 X 1 def 1 . • ιG = min π (A) QG (uv) | A ⊂ V (G) & πG (A) ≤ G 2 c def
QG (uv) =
u∈A,v∈A
♠ It is interesting to note that this parameter is actually (one of) the k.k1 versions of the spectral gap (for a proof see [32]), i.e. P |df (e)|QG (e) e P ιG = min | f : V (G) −→ R is not constant . mina u |f (u) − a|πG (u) Now, using this equality one may prove the following by the same idea of Theorem 3. Theorem 10. Let G and H be two digraphs with n = |V (G)| ≥ |V (H)| = m such that πG and πH are nowherezero stationary distributions, respectively. Then if Homv (G, H) 6= ∅ we have, M
M
Q π ιG ≤ η (G, H) Gm Hm ιH . QH πG v
Considering some graph theoretical aspects of our investigation, there are still very interesting approaches which are natural sequels to this paper. First, v e note that the study of parameters η and θ are of great importance, and consequently, finding effective methods to estimate these parameters or applying techniques (as in Theorem 4) which may help to simplify things in special cases are among the main problems. In particular, we believe it is very interesting to prove that the Coxeter graph is not Hamiltonian by this approach. Moreover, it is interesting to note that when X is a union of conjugacy classes in the Cayley graph Cay(G, X), then it is possible to compute the spectrum of the corresponding random walk from the character table of G [13, 28], which introduces a vast class of graphs that can be analysed by the methods related to the subject of this paper.
Acknowledgment Both authors wish to thank M. Shahshahani and an anonymous referee for their invaluable comments. Also, they are very grateful for the financial support of the Institute for Studies in Theoretical Physics and Mathematics (IPM).
35
References [1] M. O. Albertson and K. L. Collins, Homomorphisms of 3-chromatic graphs, Disc. Math., 54 (1985), 127–132. [2] D. Aldous and J. Fill, Preliminary version of a book on finite Markov chains, 1996. http://www.stat.berkeley.edu/users/aldous. [3] N. Alon, Eigenvalues and expanders, Combinatorica, 6 (1986), 83–96. [4] J. A. Bondy and P. Hell, A note on the star chromatic number, J. Graph Theory, 14 (1990), 479–482. ´maud, Markov Chains, Gibbs fields, Monte Carlo simulation and [5] P. Bre queues, vol. 31 of TAM, Springer–Verlag, New York, 1999. [6] M. W. Buck, Expanders and diffusers, SIAM J. Alg. Disc. Math., 7 (1986), 282–304. ´, M. Doob, and H. Sachs, Spectra of graphs, Aca[7] D. M. Cvetkovic demic Press, New York, 1980. ´, P. Rowlinson, and S. Simic ´, Eigenspaces of [8] D. M. Cvetkovic graphs, Cambridge Univ. Press, Cambridge, U.K., 1997. [9] P. Diaconis and L. Saloff-Coste, Comparison techniques for random walk on finite groups, Ann. Probab., 21 (1993), 2131–2156. [10] P. Diaconis and L. Saloff-Coste, Comparison theorems for reversible Markov chains, Ann. Appl. Probab., 3 (1993), 696–730. [11] P. Diaconis and L. Saloff-Coste, Logarithmic sobolev inequalities for finite Markov chains, Ann. Appl. Probab., 6 (1996), 695–750. [12] P. Diaconis and L. Saloff-Coste, Nash inequalities for finite Markov chains, J. Theoret. Probab., 9 (1996), 495–510. [13] P. Diaconis and M. Shahshahani, Generating a random permutation with random transpositions, Z. Wahrscheinlichkeitstheorie verw. Gebriete, 57 (1981), 159–179. [14] P. Diaconis and M. Shahshahani, Time to reach stationarity in the Bernoulli–Laplace diffusion model, SIAM J. Math. Analysis, 18 (1987), 208–218. [15] P. Diaconis and D. Stroock, Geometric bounds for eigenvalues of Markov chains, Ann. Appl. Probab., 1 (1991), 36–61.
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[16] W. D. Fellner, On minimal graphs, Theoret. Comput. Sci., 17 (1982), 103–110. [17] J. Fulman and E. L. Wilmer, Comparing eigenvalue bounds for Markov chains: when does Poincar´e beat Cheeger?, Ann. Appl. Probab., 9 (1999), 1–13. [18] C. D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993. [19] C. D. Godsil and G. Royle, Algebraic Graph Theory, vol. 207 in Graduate Texts in Mathematics, Springer-Verlag, New York, 2001. [20] W. Haemers, Eigenvalue methods in packing and covering in combinatorics, no. 106 in Math. Centre Tract, Mathematisch Centrum, Amsterdam, 1979, 15–38. [21] W. Haemers, Eigenvalue techniques in design and graph theory, PhD thesis, Eindhoven University of Technology, 1979. [22] W. Haemers, Interlacing eigenvalues and graphs, Linear Algebra Appl., 227/228 (1995), 593–616. [23] G. Hahn and C. Tardif, Graph homomorphisms: structure and symmetry, in Graph Symmetry, G. Hahn and G. Sabidussi, eds., no. 497 in NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer, Dordrecht, 1997, 107–167. ˇil, Cohomomorphisms of graphs and hyper[24] P. Hell and J. Neˇ setr graphs, Math. Nachr., 87 (1979), 53–61. ˇil, On the complexity of H–coloring, J. Combin. [25] P. Hell and J. Neˇ setr Theory (B), 48 (1990), 92–110. [26] R. Horn and C. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, U.K., 1985. ´sz, Random walks on graphs: a survey, in Combinatorics, Paul [27] L. Lova Erds is eighty, Vol. 2 (Keszthely, 1993), 353–397, Bolyai Soc. Math. Stud., 2, Jnos Bolyai Math. Soc., Budapest, 1996. [28] A. Lubotzky, Discrete groups, expanding graphs and invariant measures, vol. 125 of Progress in Mathematics, Birkh¨auser Verlag, Basel;Boston;Berlin, 1994. Appendix by J. D. Rogawski. [29] A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, Combinatorica, 8 (1988), 261–277. 37
[30] B. Mohar, A domain monotonicity theorem for graphs and Hamiltonicity, Disc. Appl. Math., 36 (1992), 169–177. [31] B. Mohar, Laplace eigenvalues of graphs – a survey, Disc. Appl. Math., 109 (1992), 171–183. [32] L. Saloff-Coste, Lectures on finite Markov chains, in Lectures on probability theory and statistics X (1996), P. Bernard, ed., no. 1665 in Lecture Notes In Mathematics, Springer–Verlag, New York, 1997, ch. 3, 304–413. [33] D. B. West, Introduction To Graph Theory, Prentice Hall, Upper Saddle River, NJ, 1996. [34] X. Zhu, Circular chromatic number, a survey, Disc. Math. 229 (1-3) (2001), 371-410.
38
GRAPH HOMOMORPHISMS THROUGH RANDOM WALKS Amir Daneshgar Department of Mathematical Sciences Sharif University of Technology P.O. Box 11365–9415, Tehran, Iran Hossein Hajiabolhassan Institute for Studies in Theoretical Physics and Mathematics (IPM) & Department of Mathematical Sciences Shahid Beheshti University P.O. Box 19834, Tehran, Iran Abstract In this paper we introduce some general necessary conditions for the existence of graph homomorphisms, which hold in both directed and undirected cases. Our method is a combination of Diaconis and Saloff– Coste comparison technique for Markov chains and a generalization of Haemers interlacing theorem. As some applications, we obtain a necessary condition for the spanning subgraph problem, which also provides a generalization of a theorem of Mohar (1992) as a necessary condition for Hamiltonicity. In particular, in the case that the range is a Cayley graph or an edge-transitive graph, we obtain theorems with a corollary about the existence of homomorphisms to cycles. This, specially, provides a proof of the fact that the Coxeter graph is a core. Also, we obtain some information about the cores of vertex–transitive graphs.
Index Words:
1
graph colouring, graph homomorphism, graph spectra, Markov chains.
Introduction
Graph homomorphisms, as natural adjacency–preserving maps between graphs, are among the most fundamental concepts in graph theory. Considering the 1 2
Correspondence should be addressed to [email protected]. This research is partially supported by the Institute for Studies in Theoretical Physics and Mathematics (IPM).
15
importance of the concept in theoretical graph theory and its profound consequences in applications, it is a natural approach, as in many other branches of mathematics, to look for no–homomorphism theorems through the study of necessary conditions which follow from the existence of such a map (it is a well-known fact that to determine the existence of such a map is usually a hard problem [25]). These conditions, in the current literature, vary from simple criteria in terms of chromatic number or odd girth of the graphs to some interesting theorems such as Albertson–Collins–Bondy–Hell no–homomorphism lemma (see Theorem B). The main objective of this paper is to introduce general no–homomorphism theorems by means of considering random walks on the corresponding graphs. To be more specific, assume the existence of an onto homomorphism σ from G to H, and consider the spectra of the corresponding random walks, which can be compared in L2 (πG ) and L2 (πH ) by means of Courant–Fischer Min-Max principle, if stationary distributions πG and πH with nonzero entries exist, respectively. The comparison technique based on this idea was first considered by P. Diaconis and L. Saloff-Coste to compare two ergodic Markov chains [9, 10, 32]. Organization of the paper is as follows. In Section 2 we go through some basic definitions and concepts from graph theory and the theory of Markov chains. Section 2.4 contains the second basic idea of the paper which has its roots in both theories, where we consider the spectrum of the quotient chain on the inverse image of σ, denoted by Gσ . The interlacing theorem for the spectrum of the quotient walk on Gσ and that of the original random walk on G is just an L2 version of the well-known contribution of Haemers [18, 20, 21, 22]. In Section 3 we apply these tools to obtain our main no–homomorphism theorems (Theorems 3–6). In Section 4 we apply our results to two important problems. Section 4.1 contains a result that can be used to test whether G is a spanning subgraph of H (Theorem 7). Also, in this section we recall a result of Saloff–Coste which can be used as another test of this type (see Corollary 4.2.6 of [32]). Also, in Section 4.2 we apply our results to obtain some information about the cores of vertex–transitive graphs, which introduces some bounds for the order and the index of regularity of such cores (Theorem 9). Moreover, we give a proof of the fact that the Coxeter graph is a core. Section 5 contains our concluding remarks.
2 2.1
Basic definitions and concepts Some preliminaries from graph theory
In this section we recall some basic concepts and notations. Also, we refer to the standard literature for what does not appear here (e.g. [33]). 16
A directed graph G (or a digraph for short) on n vertices is a structure which is determined by an ordered pair (V (G), E(G)) such that V (G) = {v1 , ..., vn } is the set of n vertices and E(G) is a family of ordered pairs (u, v), for some elements u and v of V (G), called directed edges. In this paper we only consider finite (di)graphs. Also, we identify the concepts of a simple graph (or a graph for short) and a symmetric, loopless digraph. In what follows we use the notation e = uv to denote the edge e = (u, v), where u → v and u ↔ v are used for directed and simple connections, respectively. Also, we define, →
def
E (A, B) = {uv ∈ E(G) | u ∈ A & v ∈ B & u → v}. +
Moreover, dG (v) (res. dG (v)) denotes the degree (res. out-degree) of the vertex + + v in G, and ∆G (δG ) (res. ∆G (δG )) denotes the maximum (minimum) degree (res. out-degree) of G. Kn is the complete graph on n vertices and Cn (res. → C n ) is the (res. directed) cycle on n vertices. A homomorphism f : G −→ H from a digraph G to a digraph H is a map f : V (G) −→ V (H) such that u → v implies f (u) → f (v). Similarly, a homomorphism f : G −→ H from a graph G to a graph H is a map f : V (G) −→ V (H) such that u ↔ v implies f (u) ↔ f (v). Hom(G, H), Homv (G, H) and Home (G, H) denote the sets of ordinary, onto (vertices) and onto–edges homomorphisms from G to H, respectively. A Cayley digraph Cay(G, X) of a group G with respect to the subset X ⊆ G − {0}, in which 0 is the identity element of G, has the vertex set G, and g → h if hg −1 ∈ X. It is easy to see that Cay(G, X) is connected if and only if X is a generator of G, and also Cay(G, X) is symmetric if and only if x ∈ X implies that x−1 ∈ X. Moreover, we define Cays (G, X) to be the symmetric Cayley graph Cay(G, X ∪ X −1 ). It easily follows from the definition that every Cayley graph is vertex–transitive. If H is a subgraph of G, then a retraction of G to H is a homomorphism r : G −→ H such that r(x) = x for all x ∈ V (H). A graph is a core (or a minimal graph [16]) if it does not admit a homomorphism to a proper subgraph of itself. It is easy to see [16, 24] that every graph G contains a unique (up to isomorphism) subgraph H which is a core and admits a retraction r : G −→ H. • Then usually this subgraph H is called the core of G, and is denoted by G . Also note that in this situation for each graph K, there exists a homomorphism • f : K −→ G if and only if there exists a homomorphism f : K −→ G . Theorem A. [19, 23] Let G be a vertex–transitive graph. Then, •
• G is a vertex–transitive graph. •
•
• If σ ∈ Hom(G, G ), then for all x ∈ V (G ), the inverse images σ −1 (x) (G)| have the same cardinality, namely |V|V(G • . )| 17
A digraph (graph) G is r–critical if χ(G) = r and χ(G−e) < r for each edge e ∈ E(G) (We also assume that χ(G) for any digraph G is the chromatic number of the simple graph obtained by changing all directed edges to simple ones and excluding all loops). Also, r–vertex–critical graphs are defined similarly. In [4] Bondy and Hell define ν(G, K) for two graphs G and K as the maximum number of vertices in a subgraph of G that admits a homomorphism to K; and using this they give the following generalization of a result of Albertson and Collins [1] in which µ(G, K) = |V (G)|/ν(G, K). Theorem B. [4] Let G, H, K be graphs where H is vertex–transitive. If there exists a homomorphism f : G −→ H then µ(G, K) ≤ µ(H, K).
2.2
Random walks on graphs
In this section we go through some classical known results about random walks on digraphs (e.g see [2, 5, 32]). It should be noted that all we need throughout this paper is a random walk KG on the base graph G with a stationary distribution πG such that πG (v) 6= 0 for any vertex v ∈ V (G) (by abuse of language, we call this a nowherezero stationary distribution). However, we always confine ourselves to the natural random walk on a finite graph, as an important special case, to illustrate our results. The natural random walk on a digraph G can be defined through the kernel ( 1 u→v + d (u) KG (u, v) = G 0 u 6→ v, where one can also think of KG (u, v) as the entry of a |V (G)| × |V (G)| matrix KG which is indexed by the row index u and the column index v. Note that the matrix KG is not necessarily symmetric even if the corresponding graph is a symmetric digraph, however, for regular digraphs, and of course for regular graphs as a special case, the corresponding matrix is symmetric. On the other hand, it is a simple observation that this chain is irreducible (i.e. for any u, v there exists a power k = k(u, v) such that KGk (u, v) > 0) if and only if G is a strongly connected digraph; and it is well-known that there exists a unique nowherezero stationary distribution π for such a finite Markov chain P (i.e. πKG = π, v π(v) = 1 and for each v ∈ V (G) we have π(v) 6= 0). As a matter of fact, the methods which are used in this paper are very much related to evaluation or bounding the rate of convergence to the stationary distribution (e.g. see [5, 9, 10, 11, 12, 14, 15, 27, 32]). Definition 1. define
For a random walk KG with stationary distribution πG we def
• QG (u, v) = KG (u, v)πG (u). 18
M
def
m
def
• πG = max πG (u) and πG = min πG (u). u∈V (G)
M
def
• QG =
u∈V (G)
def
m
max u6=v∈V (G)
QG (u, v) and QG =
min {QG (u, v) | QG (u, v) 6= u6=v∈V (G)
0}. Also, KG is called reversible if ∀ u, v ∈ V (G) QG (u, v) = QG (v, u).
(1) ♠
The following lemma is stated for further reference. Lemma 1. Let G be a strongly connected digraph whose natural random walk has the stationary distribution πG . Then • If G is Eulerian, +
M
πG
∆G , = |E(G)|
+
δG πG = , |E(G)|
QG = QG =
δG , 2|E(G)|
QG = QG =
m
M
m
1 . |E(G)|
• If G is simple, M
πG =
2.3
∆G , 2|E(G)|
m
πG =
M
m
1 . 2|E(G)|
Dirichlet forms and L2 (πG )
Let KG be the kernel of a random walk on V (G) with a nowherezero stationary distribution πG . We consider KG as a linear operator on L2 (πG ) of all real functions on V (G) with the following inner product def
< f, g >πG =
X
f (v)g(v)πG (v).
v∈V (G)
Consequently 12
kf kπG =
X
|f (v)|2 πG (v) ,
v∈V (G)
and it is easy to see that the adjoint of KG is defined by the kernel KG∗ (u, v) =
19
QG (v, u) , πG (u)
where it is also clear that KG is the kernel of a reversible Markov chain if and only if it is selfadjoint in L2 (πG ) i.e. KG = KG∗ . Let SG = I − 21 (KG + KG∗ ) be the additive symmetric part of KG where I is the identity operator. Then, EG (f, f )
def
= < SG (f ), f >πG =< (I − KG )f, f >πG =
X
1 2
(2)
|f (u) − f (v)|2 QG (u, v),
u,v∈V (G)
is called the Dirichlet form of KG . Moreover, all eigenvalues of SG (or those of the Dirichlet form of KG by abuse of language) are real and positive. Hence, if |V (G)| = n, then we order the eigenvalues of SG as follows G
G
G
G
0 = λ0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λn−1 . On the other hand, it is well-known that if the base graph is strongly connected (i.e. the Markov chain is irreducible) then by Perron–Frobenius theorem and G G the fact that SG is selfadjoint one may deduce 0 = λ0 < λ1 . Also, by Courant– Fischer Min-Max principle (see [26, 32]) for any 0 ≤ k < n one may write ) ( ) ( EG (f, f ) EG (f, f ) G λk = min max = max min , kf k2π kf k2π W ∈W 06=f ∈W W ∈W ⊥ 06=f ∈W k+1 k
G
G
in which def
Wk = {W ≤ L2 (πG ) | dim(W ) ≥ k},
def
Wk⊥ = {W ≤ L2 (πG ) | dim(W ⊥ ) ≤ k}.
It should be noted that in the classical theory of ergodic Markov chains the eigenvalues of the Dirichlet form are used to estimate the rate of convergence G of the chain. Note that if KG is selfadjoint then λ1 is the second smallest eigenvalue of SG = I − KG . The following theorem was used by Diaconis and Saloff-Coste in their comparison technique [9, 10, 32]. Theorem 1. [32] Let G and H be two digraphs with |V (G)| = n and |V (H)| = m ≤ n whose random walks have nowherezero stationary distributions πG and G H πH , respectively. Also, let λk and λk be the corresponding eigenvalues of SG and SH , respectively, and let T : L2 (πH ) −→ L2 (πG ) be a linear map. Then, if for some constants a > 0 and A < ∞ and for any f ∈ L2 (πH ) we have akf k2π ≤ kT (f )k2π H
then
a G λ A k
G
and EG (T (f ), T (f )) ≤ A EH (f, f ),
H
≤ λk for any 1 ≤ k ≤ m − 1.
Proof. Note that T is one to one and apply Courant–Fischer Min-Max principle (see [26, 32]). 20
It is also instructive to note that if one defines X X def def πG (f ) = πG (v)f (v) and VarπG (f ) = |f (v) − πG (f )|2 πG (v), v∈V (G)
v∈V (G)
then one may also write G
λ1
= min
EG (f, f ) | VarπG (f ) 6= 0 VarπG (f )
n o = min EG (f, f ) | kf kπG = 1, πG (f ) = 0 .
(3)
Of course, it is usually hard to compute the whole spectrum when |V (G)| is large, and one should usually obtain bounds or estimates for the eigenvalues G λk . This is a basic subject in the theory of Markov chains.
2.4
The quotient walk
The content of this section is actually a culmination of ideas from algebraic combinatorics and probability theory. Assuming a graph homomorphism σ ∈ Homv (G, H), our basic objective is to reduce the size of the state space of a random walk KG on the base graph G to a new (quotient) random walk on the base graph Gσ , in such a way that |V (Gσ )| = |V (H)|. To do this, we use a more restricted version of the technique used in Theorem 1 by choosing the linear map to be a unitary operator. This has already been done for symmetric matrices in the celebrated work of Haemers [18, 20, 21, 22]. Let σ ∈ Homv (G, H) where G and H are digraphs such that n = |V (G)| ≥ |V (H)| = m and assume that the corresponding random walks have nowherezero stationary distributions πG and πH , respectively. For any x ∈ V (H) define, def
def
Ux = σ −1 (x) and U = {Ux | x ∈ V (H)}, and note that each Ux is an independent set of vertices. Then we define the (nowherezero) probability distribution πU on the set U as X πG (v). πU (Ux ) = v∈Ux
Define the operator P : L2 (πU ) −→ L2 (πG ) through the following kernel 1 v ∈ Ux P (v, Ux ) = 0 otherwise,
(4)
and note that if we think of P as an n × m matrix, then the column which is indexed by Ux is actually the characteristic function of this set on V (G). 21
Now, it is easy to see that πU = πG P , and if we define P ∗ : L2 (πG ) −→ L2 (πU ) through the kernel def π (v) P ∗ (Ux , v) = G P (v, Ux ), πU (Ux ) then P ∗ is actually the adjoint of P i.e., < f, P (g) >πG =< P ∗ (f ), g >πU . Also, P ∗ P = IπU which shows that P is a unitary operator i.e., kP (f )kπG = kf kπU for any f ∈ L2 (πU ). σ On the other hand, consider the quotient random walk on G defined on σ def V (G ) = U through the kernel P ∗ KG P (we call this the quotient chain on the inverse image of σ). Then we have SGσ = SP ∗ K
G
P
1 = I − (P ∗ KG P + P ∗ KG∗ P ) = P ∗ SG P, 2
(5)
σ
and also the kernel of the quotient walk on G can be described explicitly through the following equation, X X QGσ (Ux , Uy ) = QG (u, v), u∈Ux
v∈Uy σ
which shows that the transition probabilities of the quotient walk on G from Ux to Uy is actually a weighted mean of the transition probabilities (e.g. number of edges when G is regular) in the original walk on G from Ux to Uy . Theorem 2. Let σ ∈ Homv (G, H) where G and H are digraphs with n = |V (G)| ≥ |V (H)| = m and nowherezero stationary distributions πG and πH , respectively. Then, ∀ 1≤k ≤m−1
G
σ G
G
λk ≤ λk ≤ λn−m+k .
Proof. This is again a direct application of Courant–Fischer Min-Max principle (see [18, 20, 21, 22, 26]) to compare the eigenvalues of SG and SGσ = P ∗ SG P (see Equation 5). Just note that in this case P is one to one since it is unitary. Moreover, note that the right hand inequality follows from the left hand one by considering the operator −SG .
3
No–homomorphism theorems
Let σ ∈ Homv (G, H) and assume that the corresponding random walks have nowherezero stationary distributions πG and πH , respectively. What we are going to show in this section is that there are relationships between the spectra 22
of SG and SH . Clearly, this will provide no–homomorphism theorems which can be used later in our special applications. σ The main setup is to consider the quotient random walk on G and then compare it with the random walk of H. This will give rise to necessary conditions in terms of the corresponding spectra (specially the spectral gap). Using these ideas we first focus on general onto homomorphisms, then we generalize our results to arbitrary homomorphisms and we also consider some special cases under some regularity or symmetry conditions.
3.1
Spectral conditions for onto homomorphisms
The comparison methods we use in this section are mainly developed by P. Diaconis and L. Saloff-Coste, however, we consider the whole spectrum to get more information [9, 10, 14, 32]. We begin by a couple of definitions. Definition 2. If σ ∈ Homv (G, H) we define, def
• Mσ =
min
→
→
{|E (σ −1 (x), σ −1 (y))| | E (σ −1 (x), σ −1 (y)) 6= ∅}.
x6=y∈V (H) σ
→
def
max |E (σ −1 (x), σ −1 (y))|.
• M =
x6=y∈V (H)
def
σ
def
• Sσ = min |σ −1 (x)| and S = max |σ −1 (x)|. x∈V (H)
x∈V (H) σ
def M Sσ
def
σ and θ = M . Sσ inf{ησ | σ ∈ Homv (G, H)} v def • η (G, H) = −∞ σ sup{θ | σ ∈ Homv (G, H)} v def • θ (G, H) = +∞
• ησ =
e
σ
Homv (G, H) 6= ∅ Homv (G, H) = ∅ Homv (G, H) 6= ∅ Homv (G, H) = ∅
e
Also, η (G, H) and θ (G, H) are define in terms of Home (G, H), similarly. ♠ The following proposition which is mainly based on Theorem 1 provides the main comparison lemma in our further applications. Proposition 1. Let σ ∈ Homv (G, H) where G and H are digraphs with n = |V (G)| ≥ |V (H)| = m and nowherezero stationary distributions πG and πH , respectively. Then for 1 ≤ k < m σ G
λk
M
M
Q π H ≤ ησ Gm Hm λk . QH πG
Also, if σ ∈ Home (G, H), then for 1 ≤ k < m σ G
λk
m
m
QG πH H ≥ θ M λk . QM π H G σ
23
Proof. First assume that σ ∈ Homv (G, H). Note that since σ is a graph homomorphism, for all x 6= y QGσ (Ux , Uy ) 6= 0 ⇒ QH (x, y) 6= 0. Also, from Equation 2 we have EGσ (f, f ) =
1 2
X
|f (y) − f (x)|2 QGσ (Ux , Uy )
x,y∈V (H)
and EH (f, f ) =
1 2
X
|f (y) − f (x)|2 QH (x, y).
x,y∈V (H)
But, since σ
M
QGσ (Ux , Uy ) ≤ M QG , we may write M
QG EGσ (f, f ) ≤ M EH (f, f ). Qm H σ
On the other hand, similarly, we have m
2
kf kπ
σ G
πG ≥ Sσ M kf k2π . H πH
Consequently, by considering the identity map ι : U −→ V (H) where U is the σ def vertex set of G , T (f ) = f ◦ ι and Theorem 1 for 1 ≤ k < m we have, σ G
λk
M
M
Q π H ≤ ησ Gm Hm λk . QH πG
Conversely, if σ ∈ Home (G, H), also for all x 6= y we have, QH (x, y) 6= 0 ⇒ QGσ (Ux , Uy ) 6= 0. Hence, using the same method we get σ G
λk
m
m
QG πH H ≥ θ λ . QM πGM k H σ
Now, we are ready to obtain one of our main no–homomorphism theorems. Theorem 3. Let G and H be two digraphs with n = |V (G)| ≥ |V (H)| = m such that πG and πH are nowherezero stationary distributions, respectively. Then 24
a ) If Homv (G, H) 6= ∅, then for all 1 ≤ k ≤ m − 1, M
M
Q π H λk ≤ η (G, H) Gm Hm λk . QH πG G
v
b ) If Home (G, H) 6= ∅, then for all 1 ≤ k ≤ m − 1, m
G
λn−m+k Proof.
m
Q π H ≥ θ (G, H) MG HM λk . QH πG e
Apply Proposition 1 and Theorem 2.
It should be noted that the full power of Theorem 2 appears in the second part, since the first part may also be obtained by a direct comparison of KG and KH through the pullback along σ. The following corollary covers many important cases since any simple graph is an Eulerian digraph as a symmetric loopless digraph. Corollary 1. Let G and H be strongly connected Eulerian digraphs with n = |V (G)| ≥ |V (H)| = m. Then a ) If Homv (G, H) 6= ∅, then for all 1 ≤ k ≤ m − 1, +
G
v
λk ≤ η (G, H)
∆H +
δG
H
λk .
b ) If Home (G, H) 6= ∅, then for all 1 ≤ k ≤ m − 1, +
G
e
λn−m+k ≥ θ (G, H) Proof.
δH
+
∆G
H
λk .
Apply Lemma 1.
Considering the case of regular graphs is instructive. Note that in this case G for an s–regular graph sλk is the kth eigenvalue of the combinatorial Laplacian. Therefore, from this point of view SG can be thought of as a generalized (normalized) version of the Laplacian of G.
3.2
Estimations and some finer tools
In this section we focus on some basic methods that can be applied to obtain v e information about the parameters η (G, H) and θ (G, H) which are needed in Theorem 3. Also, we consider methods that can be used to get rid of some of these parameters under special conditions. Needless to say, methods 25
considered in this section are just some examples of a variety of approaches that can be chosen to apply Theorem 3. Note that as first estimates, we have the following rough bounds, v
e
η (G, H) ≤ β2 (G) and α(G)−1 ≤ θ (G, H), where α(G) is the independence number of G and β2 (G) is the largest integer ε, such that there exists a subgraph G0 of G with |E(G0 )| = ε and Homv (G0 , K2 ) 6= ∅. Of course, there are a lot of known results and methods to obtain estimates for α(G) and β2 (G) (specially for simple graphs), however, getting better bounds asks for some more information about G. As an important example we mention the following simple lemma, which also shows the vast range of the cases Theorem 3 can be applied. Lemma 2. Let G and H be two digraphs, then a ) If H is vertex–critical with χ(H) = χ(G), then Hom(G, H) = Homv (G, H) and Sσ is greater than or equal to the minimum number of vertices which is needed to reduce the chromatic number of G. b ) If H is edge–critical with χ(H) = χ(G), then Hom(G, H) = Home (G, H) and Mσ is greater than or equal to the minimum number of edges which is needed to reduce the chromatic number of G. In the next example we consider the set Hom(P10 , C5 ), of all homomorphisms from the Petersen graph P10 to the 5–cycle C5 . Although, it will easily follow from our next results that Hom(P10 , C5 ) = ∅, but we follow the details of the next example to show how one may estimate the necessary parameters. Example 1. Let us consider a homomorphism σ ∈ Hom(P10 , C5 ). First, note that since P10 contains two disjoint copies of C5 , we can deduce that σ S = 2 (note that this also follows from the fact that P10 is vertex–transitive and that C5 is a core). Since C5 is critical, the homomorphism should be in Home (P10 , C5 ), and by Lemma 2(b) we have Mσ ≥ 3. Hence, applying Theorem 3(b) for k = 3 shows that we should have 1.5 ≤
Mσ e σ ≤ θ (G, H) ≤ 1.38, S
which is a contradiction. Hence, Hom(P10 , C5 ) = Home (P10 , C5 ) = ∅. (Note that Theorem 3(a) is not applicable in this case.) ♣ Example 2. In this example we consider the digraphs G = Cay(Z22 , {2, 3, 4, 5, 6, 7, 8, 9}) and H = Cay(Z11 , {1, 2, 3}), 26
where we consider the existence of a homomorphism σ in Hom(G, H). First, we note it is not hard to check that χ(G) = χ(H) = 6, and that H is a 6–critical digraph; which shows that Hom(G, H) = Home (G, H). Also, it can be checked that we need to delete at least two vertices of G to reduce its chromatic number to 5. Hence, by Lemma 2 we have Sσ ≥ 2, and consequently, since any homomorphism should be onto the vertices of H we σ σ have Sσ = S = 2. Moreover, trivially, Mσ ≥ 1 and M ≤ 4. Now, applying Theorem 3(a) for k = 1 we have, σ
M 3.76 ≤ η (G, H) ≤ ≤ 2, Sσ v
which is a contradiction. Therefore, Hom(G, H) = Home (G, H) = ∅. First, note that Theorem 3(b) is not applicable in this case. Also, note that since both G and H are regular digraphs, the above inequality also holds for the ˜ = Cays (Z22 , {2, 3, 4, 5, 6, 7, 8, 9}) and H ˜ = Cays (Z11 , {1, 2, 3}). simple graphs G e ˜ ˜ ˜ ˜ Hence, we have Hom(G, H) = Hom (G, H) = ∅. ♣ As it is clear, one of the main problems in applying Theorem 3 is to obtain v e good estimates for parameters η and θ . The following theorem presents one σ of the techniques that may be used to simplify this by excluding Sσ and S from computations. Theorem 4. Let G and H be two digraphs with n = |V (G)| ≥ |V (H)| = m, where πG and πH are nowherezero stationary distributions, and H = Cay(V (H), X) is a Cayley graph in which X is closed under conjugation. Then if σ ∈ Home (G, H), for all 1 ≤ k ≤ m − 1 we have, m m QGH πH H mMσ GH λk , λm(n−1)+k ≥ 1 + M n QM πGH H where GH is the cartesian product of the graphs G and H. Proof. Let V (G) = {v1 , . . . , vn }, V (H) = {x1 , . . . , xm }, and consider a homomorphism σ ∈ Home (G, H). We define a map σ ˜ : V (GH) −→ V (H) as follows, σ ˜ ((vi , xj )) = σ(vi )xj i = 1, . . . , n and j = 1, . . . , m. First, we show that σ ˜ is a homomorphism. Let xj → xk in H. Consequently, xk x−1 ∈ X and (v , xj ) → (vi , xk ) in GH. But, since X is closed under i j conjugation we have, (σ(vi )xk )(σ(vi )xj )−1 = σ(vi )(xk x−1 )σ(vi )−1 ∈ X, j which shows that σ(vi )xj → σ(vi )xk in H. On the other hand, let vj → vk in G and consequently, (vj , xi ) → (vk , xi ) in GH. Again, since σ is a homomorphism we have, (σ(vk )xi )(σ(vj )xi )−1 = σ(vk )σ(vj )−1 ∈ X, 27
which shows that σ(vj )xi → σ(vk )xi in H. These show that σ ˜ is a homomorphism of graphs. σ ˜ Now, we show that S = Sσ˜ = n. To see this, consider a fixed vertex xt ∈ V (H). Then, if (vi , xj ) ∈ σ ˜ −1 (xt ), we may compute xj as xj = σ(vi )−1 xt , since V (H) is a group. Hence, |˜ σ −1 (xt )| ≥ n. However, since σ ˜ ∈ Homv (GH, H) and |GH| = mn we have |˜ σ −1 (xt )| = n for any t = 1, . . . , m. This shows σ ˜ that S = Sσ˜ = n. To estimate Mσ˜ fix an edge xt → xs in H. Then, for any 1 ≤ j ≤ m we have, σ ˜ −1 (xj ) = {(vi , σ(vi )−1 xj ) | vi ∈ V (G)}. We consider two cases, • The first components are equal. In this case vi is fixed. Also, since X is closed under conjugation we have σ(vi )−1 xt → σ(vi )−1 xs which shows that (vi , σ(vi )−1 xt ) → (vi , σ(vi )−1 xs ) in GH. This shows that we have n edges between σ ˜ −1 (xt ) and σ ˜ −1 (xs ) of this type. • The second components are equal. In this case we want to estimate the number of edges of the type (vi , σ(vi )−1 xt ) → (vj , σ(vj )−1 xs ) where σ(vi )−1 xt = σ(vj )−1 xs
(i 6= j).
This means that we should have σ(vj )σ(vi )−1 = xs x−1 ∈X t
(i 6= j) (∗).
Now if σ ∈ Home (G, H), then there are vertices vi and vj such that vi → vj in G, σ(vi ) = xt and σ(vj ) = xs ; which means that (∗) has at least one solution. Consequently, there are at least Mσ solutions for (∗) in this way which come from edges in the inverse image of σ. On the other hand, for each fixed x ∈ V (H) we have xt x → xs x in H; and again by considering the inverse image of σ we find another bunch of Mσ solutions for (∗). This shows that there are at least mMσ edges between σ ˜ −1 (xt ) and σ ˜ −1 (xs ) of this type when σ ∈ Home (G, H). These show that Mσ˜ ≥ n+mMσ and the rest of proof follows from Theorem 3. First, note that for the cartesian product as well as many other different products, the eigenvalues can be computed explicitly in terms of the eigenvalues of the factors (for the definition of a NEPS of graphs and its eigenvalues see [7, 8]). Moreover, the proof of Theorem 4 can be sharpened when H is a (directed) cycle as follows, in which there is no need to have an estimate for Mσ . 28
Corollary 2. Let G be a digraph such that such that πG is a nowherezero → stationary distributions, n = |V (G)| and ε = |E(G)|. Also, let H ∈ {Cm , C m } with m ≤ n. Then if σ ∈ Home (G, H), for all 1 ≤ k ≤ m − 1 we have, m
GH
λm(n−1)+k ≥ Proof. have
m
ε QGH πH H 1+ λk . M n QM πGH H
Note that in cycles as Cayley graphs for any fixed edge x → y we {xz → yz | z ∈ V } = E.
So far we have focused on onto homomorphisms. Now, we introduce another variant which gives rise to a necessary condition for the existence of a (not necessarily onto) homomorphism that can be used to prove many different and general no-homomorphism theorems. In what follows we propose two such applications. Theorem 5. Let G and H be two digraphs such that |V (G)| = n, |V (H)| = m and πG is a nowherezero stationary distribution. Also, assume that the group of automorphisms of H, Aut(H), acts transitively on both V (H) and E(H). If σ ∈ Hom(G, H) then, m
G
λn−1
2|E(G)|QG H ≥ λm−1 . nπGM G
H
Specially, if G is s-regular then λn−1 ≥ λm−1 . Proof.
Let |Aut(H)| = t, Aut(H) = {ζi | i = 1, . . . , t} and define, t
def
˜ = G
[
Gi ,
i=1
˜ such as G , is an isomorphic copy of where each connected component of G, i G. Also, define the homomorphism σ ˜ such that its restriction to Gi is ζi ◦ σ. σ |V (G)| |E(G)| × t and S = |V × t. It is easy to see that σ ˜ ∈ Home (G, H), Mσ = |E(H)| (H)| Moreover, if vi ∈ V (Gi ) is the vertex corresponding to the vertex v ∈ V (G), def π (v) then π ˜G˜ (vi ) = Gt defines a nowherezero stationary distribution π ˜G˜ for the ˜ natural random walk of G. The rest of the proof follows from Theorem 3(b).
29
It is easy to see that this theorem proves the following well-known lower bound for the chromatic number of an s-regular graph (in which µn is the least eigenvalue of the adjacency matrix of G), χ(G) ≥ 1 −
s . µn
Also, note that using the same method one may prove other different variants of this theorem. In what follows we mention one of these cases for its possible applications in graph decompositions. Theorem 6. Let G and H be two digraphs such that |V (G)| = n, |V (H)| = m and πG is a nowherezero stationary distribution. Also, assume that H is vertextransitive and r-regular. If σ ∈ Home (G, H) then, m
G
λn−1
rmMσ QG H ≥ λm−1 . nπGM
First, note that the number of eigenvalues that can be compared in theorems ˜ like this, heavily depends on the number of copies used to form the graph G. ˜ to H with using just Hence, when one can construct homomorphisms from G a small number of copies of G, then one may be able to use more inequalities. Also, note that the same method can be used in the case of onto (vertices) homomorphisms if we know the number of copies explicitly. We would like to emphasize that this type of general homomorphism theorems can be quite useful in the study of graph decompositions and the study of other concepts in graph theory which are defined in terms of graph homomorphisms. In the next section we just mention a simple case which is related to the spanning subgraph problem.
4 4.1
Some applications Spanning subgraphs and Hamiltonian cycles
An important case is when G is a spanning subgraph of H. Note that in this σ case considering σ as the identity homomorphism we have Sσ = M = 1. Hence, we have the following as a corollary of Theorem 3. Theorem 7. Let G and H be two digraphs with |V (G)| = |V (H)| = n such that πG and πH are nowherezero stationary distributions, respectively. If there exists 1 ≤ k < n such that M
M
Q π H λk > Gm Hm λk , QH πG G
then G is not a spanning subgraph of H. 30
Example 3. To show that the Petersen graph is not Hamiltonian, we should P show that Homv (C10 , P10 ) = ∅, which follows from Theorem 7 for λ5 10 . ♣ Note that this theorem generalizes a theorem of Mohar [30, 31]. Example 4. In this example we consider the existence of homomorphisms to circular graphs Gnd , which are defined on the vertex set V (Gnd ) = {0, . . . , n − 1} in which i ↔ j if d ≤ |i − j| ≤ n − d (for applications and more results see [34]). As an special case we note that the Kneser graph K(2, 6), is not a spanning K(2,6) subgraph of G15 , by applying Theorem 7 for λ10 . ♣ 4 Getting a similar result by comparison from the other way round is not easy in general, however, when H and G are regular graphs we have the following proposition which is essentially the Corollary 4.2.6 of [32]. We add the proof for completeness and to emphasize the application of flows to the subject. Theorem 8. Let G and H be two regular graphs with |V (G)| = |V (H)| = n, where G is edge-transitive and πG and πH are nowherezero stationary distributions, respectively. If G is a spanning subgraph of H, then |E(H)| G H λk ≥ X λk , dG (x, y)2
1 ≤ k ≤ n − 1.
x,y∈V (G)
Proof. Let ΓG (ΓG (x, y)) be the set of all paths in G (from x to y), and also let GG (x, y) ⊆ ΓG (x, y) be the set of all geodesics (shortest paths) from x to y. We use a comparison through the identity map from L2 (πG ) to L2 (πH ). But, first we note that • For each xy ∈ E(H) and each γ ∈ GG (x, y) by Cauchy–Schwarz inequality we have X |f (y) − f (x)|2 ≤ |γ| |f (v) − f (u)|2 . uv∈γ
• If we define Φ on ΓG as ( def
Φ(γ) =
QH (x,y) #GG (x,y)
0
γ ∈ GG (x, y) otherwise,
Then it is clear that (i.e. Φ is a flow), X ∀ xy ∈ E(H), x 6= y, γ∈Γ (x,y) G
31
Φ(γ) = QH (x, y).
Therefore, X
|f (y) − f (x)|2 QH (x, y) ≤
|γ|
X
|f (v) − f (u)|2 Φ(γ).
uv∈γ
γ∈Γ (x,y) G
Hence, EH (f, f ) =
X
1 2
|f (y) − f (x)|2 QH (x, y)
x,y∈V (H)
≤
1 2
X
X
|γ|
X
|f (v) − f (u)|2 Φ(γ)
uv∈γ
x,y∈V (H) γ∈Γ (x,y) G
def |γ|Φ(γ) EG (f, f ) = M EG (f, f ),
X
≤ (QG (x0 , y0 ))−1
x0 y0 ∈γ∈Γ G
for some fixed edge x0 y0 ∈ E(G). On the other hand, X M = (QG (x0 , y0 ))−1 |γ|Φ(γ) x0 y0 ∈γ∈Γ G
=2
X
X
e∈E(G)
≤2
X
|γ|Φ(γ)
e∈γ∈Γ G
X
X
e∈E(G) x,y∈V (G)
≤
1 |E(H)|
X
e∈γ∈G (x,y) G
dG (x, y)QH (x, y) #GG (x, y)
dG (x, y)2 .
x,y∈V (G)
Also, for the corresponding norms we have, X 1 kf kπH = |f (x)|2 = kf kπG . |V (G)| x∈V (G)
Consequently, by Theorem 1 we have X 1 H G λk ≤ dG (x, y)2 λk |E(H)|
∀ k ∈ {1, . . . , n − 1},
x,y∈V (G)
which completes the proof.
4.2
On vertex transitive cores
In what follows we are going to apply our results of Section 3 to obtain some information about cores. Let G be a vertex–transitive graph on n vertices, and let def O(n) = {m ∈ N | m is odd and divides n}. 32
First, we note that by Theorem A, if G is 3–regular, and there is no homomorphism from G to the cycle Cm for all m ∈ O(n), then G is a core. This can be effectively verified using Theorems 1 and 4. To show this, first we may note that Example 1 can be considered as an application of Corollary 1 to prove that the Petersen graph is a core. This also σ could be verified by Corollary 2 without estimating the parameters S and Mσ . We consider the same idea in the following example which is about the Coxeter graph. Example 5. By applying Corollary 2 to Hom(G, C7 ), where G is the Coxeter graph and k = 6, we get 2 3 1.843 ≥ (1 + ) × × 1.9 = 1.9, 2 5 which shows that Hom(G, C7 ) = ∅. This proves that the Coxeter graph is a core. It is also interesting to note that the only other proof of this, that the authors are aware of, uses the fact that any connected 2–arc transitive non–bipartite graph is a core [19, 23]. ♣ Theorem 9. Let G be a simple s–regular vertex–transitive graph with |V (G)| = n = rm whose core is a t–regular subgraph, through the homomorphism σ, with • r σ |V (G )| = m. Also, let M ≥ M and Mr ≤ Mσ . Then, t G r G ∀1≤k<m rλk ≤ M (λn−m+k + − 1), s and t G G ∀1≤k<m rλn−m+k ≥ Mr (λk + − 1). s Proof. First, note that if H is an induced t–regular subgraph of an s–regular graph G, then by considering the adjacency matrices and the interlacing lemma we have, ∀ 1≤k ≤m−1
G
H
G
s(1 − λn−m+k ) ≤ t(1 − λk ) ≤ s(1 − λk ),
in which |V (H)| = m ≤ |V (G)| = n. • Now, let |V (G )| = m and |V (G)| = rm for some integer r ≥ 1. Hence, • σ considering the corresponding homomorphism σ : G −→ G we have M ≤ r M , Sσ = r; and, consequently, by Corollary 1, r
r
tM G• tM s G λk ≤ (1 − (1 − λn−m+k )); sr sr t which proves the first inequality. The second inequality is also proved similarly. ∀ 1≤k ≤m−1
G
λk ≤
Note that the theorem essentially shows that m, the order of the core, can not be arbitrary small. Also, it is clear that the theorem provides upper and lower bounds for t, the degree of regularity of the core. 33
5
Concluding remarks
It is instructive to add some notes on the whole setup we have introduced so far and point out some important aspects that can be considered in forthcoming research. First, we wish to emphasize that the method introduced in this paper works when • There are two (not necessarily natural) random walks KG and KH such that their base graphs are G and H respectively, and there are (not necessarily unique) stationary distributions πG and πG which are nonzero on each vertex. • The operator SG is a well-behaved selfadjoint operator obtained from KG ∗ def (e.g. one may consider many other variations such as SG = I − KG KG ). • One may use different perturbation methods, such as adding loops, to define various random walks and use Theorem 3 more effectively (e.g. note that by adding loops properly, one may change a nonregular graph to a regular one). It is evident from our approach that any kind of information about the spectrum of the natural random walk of a graph has important consequences on what we may deduce about its homomorphic images or the graphs that have homomorphisms to it. This, in a special case, may draw ones attention to expanders and Ramanujan graphs as an important class of random–like graphs with a relatively large spectral gap. Even it is quite interesting to consider families of s–regular√graphs for which the spectral gap (in the classical sense) is greater than s−2 s − 1+ for some > 0 (note that by Alon–Boppana theorem there are only a finite number of such graphs for a fixed s (see [3, 6, 28, 29] and references therein)). Also, one should note that, since it is not always easy to compute the whole spectrum, it is usual to focus on the spectral gap, λ1 , or some other parameters which are mainly related to the rate of convergence of the chain, like the log–Sobolev or the Cheeger constants. It is interesting to note that one may also use the techniques of the previous sections to obtains some partial results concerning these parameters too. We also note that the role of these parameters can be quite different from the spectral gap, which, in a way, justifies their consideration as a separate subject (for more on these subjects see [11, 12, 15, 17, 32]). For instance, we may consider one special type of the Cheeger constant as follows. Definition 3. Let G be a digraph whose natural random walk has the nowherezero stationary distribution πG . Then 34
• For any edge e = uv we define 1 def (QG (u, v) + QG (v, u)) and df (uv) = f (v) − f (u). 2 X 1 def 1 . • ιG = min π (A) QG (uv) | A ⊂ V (G) & πG (A) ≤ G 2 c def
QG (uv) =
u∈A,v∈A
♠ It is interesting to note that this parameter is actually (one of) the k.k1 versions of the spectral gap (for a proof see [32]), i.e. P |df (e)|QG (e) e P ιG = min | f : V (G) −→ R is not constant . mina u |f (u) − a|πG (u) Now, using this equality one may prove the following by the same idea of Theorem 3. Theorem 10. Let G and H be two digraphs with n = |V (G)| ≥ |V (H)| = m such that πG and πH are nowherezero stationary distributions, respectively. Then if Homv (G, H) 6= ∅ we have, M
M
Q π ιG ≤ η (G, H) Gm Hm ιH . QH πG v
Considering some graph theoretical aspects of our investigation, there are still very interesting approaches which are natural sequels to this paper. First, v e note that the study of parameters η and θ are of great importance, and consequently, finding effective methods to estimate these parameters or applying techniques (as in Theorem 4) which may help to simplify things in special cases are among the main problems. In particular, we believe it is very interesting to prove that the Coxeter graph is not Hamiltonian by this approach. Moreover, it is interesting to note that when X is a union of conjugacy classes in the Cayley graph Cay(G, X), then it is possible to compute the spectrum of the corresponding random walk from the character table of G [13, 28], which introduces a vast class of graphs that can be analysed by the methods related to the subject of this paper.
Acknowledgment Both authors wish to thank M. Shahshahani and an anonymous referee for their invaluable comments. Also, they are very grateful for the financial support of the Institute for Studies in Theoretical Physics and Mathematics (IPM).
35
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