Gaddum-type inequality for the hyper-Wiener ... - Semantic Scholar
Theoretical Computer Science 471 (2013) 74–83
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Nordhaus–Gaddum-type inequality for the hyper-Wiener index of graphs when decomposing into three parts Guifu Su a,∗ , Liming Xiong a,b , Yi Sun c , Daobin Li d a
School of Mathematics, Beijing Institute of Technology, Beijing, 100081, PR China
b
Department of Mathematics, Jiangxi University, Jiangxi, 330026, PR China
c
College of Mathematics and System Science, Xinjiang University, Xinjiang, 830046, PR China
d
School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing, 100083, PR China
article
abstract
info
Article history: Received 1 May 2012 Received in revised form 25 September 2012 Accepted 11 October 2012 Communicated by D.-Z. Du Keywords: Wiener index Hyper-Wiener index k-decomposition Nordhaus–Gaddum-type inequality
Let k ≥ 2 be an integer, a k-decomposition (G1 , G2 , · · · , Gk ) of a graph G is a partition of its edge set to form k spanning subgraphs G1 , G2 , . . . , Gk . The hyper-Wiener index WW is one of the recently conceived distance-based graph invariants (Randi 1993 [15]): WW = WW (G) := 12 W (G) + 12 W2 (G), where W is the Wiener index (Wiener 1947 [18]) and W2 is the sum of squares of distance of all pairs of vertices in G. In this paper, we investigate the Nordhaus–Gaddum-type inequality of a 3-decomposition of Kn for the hyper-Wiener index: 7
n 2
≤ WW (G1 ) + WW (G2 ) + WW (G3 ) ≤ 2
n+2 4
+
n 2
+ 4(n − 1).
The corresponding extremal graphs are characterized. Published by Elsevier B.V.
1. Introduction Throughout the paper all graphs considered are finite and simple. Let G = (V (G), E (G)) be a graph, we use |V | = |V (G)| and e(G) = |E (G)| to denote its order and size, respectively. The degree of a vertex u is the number of edges incident with it in G, denoted by degG (u), or deg(u) when no confusion is possible. As usual, △(G) and δ(G) denote the maximum and the minimum degree of G, respectively. The distance distG (u, v) between vertices u and v is the length of the shortest path in G connecting them. The diameter diam(G) of a graph G is the maximal distance between any two vertices. The complement of G, denoted by G, is a simple graph on the same set of vertices V in which two vertices are adjacent if and only if they are not adjacent in G. Let k ≥ 2 be an integer, a k-decomposition (G1 , G2 , . . . , Gk ) of a graph G is a partition of its edge set to form k spanning subgraphs G1 , G2 , . . . , Gk , and each Gi is said to be a cell. In other words, each Gi has the same vertices as G, and each edge of G belongs to exactly one of G1 , G2 , . . . , Gk . In particular, (G1 , G2 ) is a 2-decomposition of Kn if and only if G1 is the complement of G2 . Other terminology and notations needed will be introduced as it naturally occurs in the following and we use [2] for those not defined here. The Wiener index W of a connected graph G is the sum of distance of all pairs of vertices in it [18]: W = W (G) :=
distG (u, v),
{u,v}∈V (G)
∗ Correspondence to: Computational Systems Biology Lab, Department of Biochemistry and Molecular Biology, The University of Georgia, GA, USA. Tel.: +86 01013717961438. E-mail address: [email protected] (G. Su). 0304-3975/$ – see front matter. Published by Elsevier B.V. doi:10.1016/j.tcs.2012.10.049
G. Su et al. / Theoretical Computer Science 471 (2013) 74–83
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which was introduced by the chemist Wiener in 1947. Its chemical applications and mathematical properties were raised in [3,13,16]. The hyper-Wiener index WW is one of the recently conceived distance-based graph invariants [15]: WW = WW (G) :=
1 2
W (G) +
1 2
W2 (G),
2 where W2 (G) = {u,v}∈V (G) distG (u, v). This index was introduced by Randić and used as a structure-descriptor for predicting physico-chemical properties of organic compounds, often those significant for pharmacology, agriculture, environmental protection and so on. It rapidly gained popularity and numerous results on it were stated in [6–9,11]. Let l be a positive integer not less than 1, and β be a parameter of graph G, to determine the extremal (maximum or minimum) value of
l
β(Gi ) : (G1 , G2 , . . . , Gl ) is a l-decomposition of G
i=1
is a fundamental problem in graph theory. The particular case when G = Kn attracts much attention on various graph parameters. It was Nordhaus and Gaddum [14] who first initiated such kind of research on chromatic number of graphs for the case when l = 2 and G = Kn . They proved that:
√
2 n ≤ χ (G) + χ(G) ≤ n + 1
and
n ≤ χ (G) · χ (G) ≤
n+1 2
2
,
where χ denotes the chromatic number of graph G. Zhang and her co-workers [19] showed that: Theorem A (Zhang et al. [19]). Let (G1 , G2 ) be a 2-decomposition of Kn such that each cell Gi is connected. Then for any sufficiently large n, we have
4
n
2
n+4 . ≤ WW (G1 ) + WW (G2 ) ≤ 2 4
The lower and the upper bounds are sharp. Later, Li et al. paid their attention to the following for the diameter in [10]. They said: Theorem B (Li et al. [10]). Let (G1 , G2 , G3 ) be a 3-decomposition of Kn such that each cell Gi is connected. Then for any sufficiently large n, we have
5
n
2
≤ W (G1 ) + W (G2 ) + W (G3 ) ≤
n3 − n 3
+
n
2
+ 2(n − 1).
The lower and the upper bounds are sharp. Motivated by Theorems A and B, in this paper, we consider the Nordhaus–Gaddum-type inequality of a 3-decomposition of Kn for the hyper-Wiener index. The corresponding extremal graphs are characterized. 2. Preliminary lemmas In this section we list or prove some lemmas as basic but necessary preliminaries, which will be used in the subsequent proofs. Lemma 2.1 (An et al. [1]). Let G be a simple graph with order n. If δ(G) ≥
n 2
, then diam(G) ≤ 2.
− 1. Lemma 2.2 (Erdös et al. [4]). Let G be a connected graph with order n. Then diam(G) ≤ δ(G3n )+1 G
:G
Let us introduce some useful notations. A splice a1 :a 2 of two connected graphs G1 and G2 is a graph obtained by 1 2 identifying the vertices a1 ∈ V1 and a2 ∈ V2 . K :P Let n and d be two integers with n > d, we denote Tn,d = 1b,d :bn−d the graph obtained from identifying a leaf b1 of the 1 2 star K1,d with a leaf b2 of the path Pn−d . It is trivial to see that the graph Tn,d has maximum degree d and order n. In particular, Tn,d = Pn when d = 2. By some calculations, we have W (Tn,d ) = (d − 1)(d − 2) +
(d − 1)(n − d + 1)(n − d + 2) 2
+
(n − d + 1)3 − (n − d + 1) 6
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and W2 (Tn,d ) = 2(d − 1)(d − 2) +
(d − 1)(n − d + 1)(n − d + 2)(2n − 2d + 3) 6
+
(n − d)(n − d + 2)(n − d + 1)2 12
.
For simplicity, we denote Φ1 (n, d) = W (Tn,d ) and Φ2 (n, d) = W2 (Tn,d ) in sequel. Then we have the following consequence. Lemma 2.3. For 2 ≤ d ≤ n − 1, Φ (n, d) =
1 Φ 2 1
(n, d) + 21 Φ2 (n, d) is a decreasing function on d.
Proof. This result can be verified directly, so we omit the proof here. Lemma 2.4. Let n be a fixed positive integer and c a real number with n > 2c > 0. Then the function
σ (x 1 , x 2 ) =
2 xi (n − xi + 2)2 i =1
+
2
takes the maximum value at x1 = x2 =
c 2
(n − xi + 2)3
+
6
xi (n − xi + 2)3 3
+
(n − xi + 2)4
12
if x1 + x2 ≥ c .
Proof. For simplicity, let f (x1 , x2 ) =
2 xi (n − xi + 2)2 i=1
2
+
(n − xi + 2)3
+
(n − xi + 2)4
6
and g (x1 , x2 ) =
2 xi (n − xi + 2)3 i=1
3
12
.
Li et al. [10] proved that the function f (x1 , x2 ) takes the maximum value at (x1 , x2 ) = ( 2c , 2c ). Here we only need to consider the problem: maximize g (x1 , x2 ) subject to x1 + x2 = c . By means of Lagrange multiplier [17], we show that the maximum value of g (x1 , x2 ) is also attained at (x1 , x2 ) = ( 2c , 2c ). Put R(x1 , x2 , λ) = g (x1 , x2 ) + λ(x1 + x2 − c ). Set the derivative dR = 0, which yields the following system of equations:
∂R = −x1 (n − x1 + 2)2 + λ = 0. ∂ x1 ∂R = −x2 (n − x2 + 2)2 + λ = 0. ∂ x2 ∂R = x1 + x2 − c = 0. ∂λ Combining the above equations, we have x1 = x2 =
c 2
by easy calculations.
Note that the function g (x1 , x2 ) takes the maximum value when x1 = x2 = where A=
c 2
, since △ = B2 − AC < 0 and A < 0 [17],
∂ 2R 3 = −(n + 2 − 3x1 )(n + 2 − x1 ) = − n + 2 − c (n + 2 − c ) < 0. 2 2 ∂ x1
∂ 2R ∂ 2G = = 0. ∂ x1 x2 ∂ x2 x1 ∂ 2R 3 C = = −(n + 2 − 3x2 )(n + 2 − x2 ) = − n + 2 − c (n + 2 − c ) < 0. 2 2 ∂ x2 B=
Hence, σ (x1 , x2 ) = f (x1 , x2 ) + g (x1 , x2 ) takes the maximum value at x1 = x2 = This completes the proof of Lemma 2.4.
c 2
subject to x1 + x2 ≥ c .
Lemma 2.5 (An et al. [1]). Let Kn be the complete graph with order n and k ≥ 2 any fixed integer. Then for any sufficiently large n with respect to k, there exists a k-decomposition (G1 , G2 , . . . , Gk ) of Kn such that diam(Gi ) = 2 for each i = 1, 2, . . . , k. Lemma 2.6. Let G be a connected non-complete graph and e be its non-cut-edge. Then WW (G − e) + 2e(G − e) ≥ WW (G) + 2e(G).
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Proof. Let e = uv ∈ E (G), by some computation W (G) =
distG (u, x) +
x̸=v
distG (v, y) + distG (u, v) + distG (v, u) +
distG (p, q)
p̸=u,q̸=v
y̸=u
and W (G − e) =
distG−e (u, x) +
x̸=v
distG−e (v, y) + distG−e (u, v) + distG−e (v, u) +
distG−e (p, q),
p̸=u,q̸=v
y̸=u
which implies that W (G − e) > W (G) + 1. By analogous argument we get W2 (G − e) > W2 (G) + 1. Then W (G − e) + 2e(G) > W (G) + 2e(G) + 1
⇒ W (G − e) + 2e(G) − 1 ≥ W (G) + 2e(G) + 1 ⇒ W (G − e) + 2e(G − e) ≥ W (G) + 2e(G). In a similar way, we have W2 (G − e) + 2e(G − e) ≥ W2 (G) + 2e(G). Therefore WW (G − e) + 2e(G − e) =
= ≥
1 2 1 2 1 2 1
W (G − e) +
1 2
W2 (G − e) + 2e(G − e) 1
[W (G − e) + 2e(G − e)] + [W2 (G − e) + 2e(G − e)] 2
1
[W (G) + 2e(G)] + [W2 (G) + 2e(G)] 2
1 W (G) + W2 (G) + 2e(G) 2 2 = WW (G) + 2e(G).
=
This completes the proof of Lemma 2.6. Lemma 2.7. Let (G1 , G2 , G3 ) be a 3-decomposition of Kn such that each cell Gi is connected. Then there exists at most one cell Gi with δ(Gi ) ≥ 2n . Proof. By contradiction, assume there are at least two cells of {G1 , G2 , G3 }, say G1 and G2 , such that δ(Gi ) ≥ By the Handshaking Lemma, we have 2e(G1 ) + 2e(G2 ) =
u∈V (G1 )
degG1 (u) +
u∈V (G2 )
degG2 (u) ≥
1 2
n2 +
1 2
n 2
for i = 1, 2.
n2 = n2 > n(n − 1) = 2e(Kn ),
which is a contradiction. This completes the proof of Lemma 2.7. 3. Upper bounds on hyper-Wiener index Let Sn denote the star with n vertices. Gutman and his co-workers determined trees with minimal and maximal hyperWiener indices in [5]. Theorem 3.1 (Gutman et al. [5]). Let T be a tree with order n. Then WW (Sn ) ≤ WW (T ) ≤ WW (Pn ). Theorem 3.2 (Liu et al. [12]). Let G′ be a connected spanning subgraph of G. Then WW (G) ≤ WW (G′ ). Combining the above theorems, we get: Corollary 3.3. Let G be a connected graph with order n. Then WW (G) ≤
, with equality if and only if G = Pn . 4 n+2 Proof. Note that WW (Pn ) = 4 . Let T be a spanning tree of G, then WW (G) ≤ WW (T ) by Theorem 3.2. By Theorem 3.1, we have WW (G) ≤ WW (Pn ), as desired. n+2
Let α ≥ 1 be an integer, the α -transmission of a vertex u, denoted by Trα (u : G), is the sum of distances from it to all the other vertices in graph G, i.e., Trα (u) = Trα (u : G) =
distαG (u, v).
v∈V
Theorem 3.4. Let Ti be a tree with order ni and a specified vertex xi for i = 1, 2. Then WW ( and xi is a leaf of Ti .
T1 :T2 x1 :x2
) is maximized only if Ti = Pni
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G. Su et al. / Theoretical Computer Science 471 (2013) 74–83
Fig. 1. The transformation T1 → T2 increases the WW -value.
Proof. For convenience, let V (T1 ) = {y1 , y2 , . . . , yi0 = x1 , . . . , yn1 }, V (T2 ) = {z1 , z2 , . . . , zj0 = x2 , · · · , zn2 } and G=
T1 :T2 . It is easily seen that x1 :x2
W (G) = W (T1 ) + W (T2 ) + (n2 − 1)Tr(x1 : T1 ) + (n1 − 1)Tr(x2 : T2 ). On the other hand, dist2G (yi , zj ) = [distT1 (x1 , yi ) + distT2 (x1 , zj )]2 = dist2T1 (x1 , yi ) + dist2T2 (x1 , zj ) + 2distT1 (x1 , yi ) · distT2 (x1 , zj ). Hence, the contribution from pairs of vertices yi and zj to W2 (G) equals to the sum: n1 n2
dist2G (yi , zj ) =
i=1 j=1
n1 n2 [distT1 (x1 , yi ) + distT2 (x1 , zj )]2 i=1 j=1
=
n1 n2 i=1 j=1
dist2T1 (x1 , yi ) +
n1 n2
dist2T2 (x1 , zj ) + 2
i=1 j=1
n1 n2
distT1 (x1 , yi ) · distT2 (x1 , zj )
i=1 j=1
= n2 Tr2 (x1 : T1 ) + n1 Tr2 (x2 : T2 ) + 2Tr(x1 : T1 ) · Tr(x2 : T2 ). Thus, we have W2 (G) = W2 (T1 ) + W2 (T2 ) + (n2 − 1)Tr2 (x1 : T1 ) + (n1 − 1)Tr2 (x2 : T2 ) + 2Tr(x1 : T1 ) · Tr(x2 : T2 ). Hence, WW (G) = WW (T1 ) + WW (T2 ) + Tr(x1 : T1 ) · Tr(x2 : T2 ) 1 1 + (n2 − 1)[Tr2 (x1 : T1 ) + Tr(x1 : T1 )] + (n1 − 1)[Tr2 (x2 : T2 ) + Tr(x2 : T2 )]. 2 2 By Corollary 3.3, we complete the proof as desired. Next, we will introduce two auxiliary transformations, which increase the WW -value of graphs. Transformation I. Let T be a tree and v a vertex with maximum degree in T . Let T0 be a component of T − v. By replacing T0 with a path with the same order and the same pendent vertex as T0 under the definition of splice, we obtain a new tree ′ ′ T . By Theorem 3.4, we confirm that the transformation T → T increases the hyper-Wiener index. Transformation II. Consider the trees T1 and T2 depicted in Fig. 1. These two trees differ only in the position of a terminal vertex: in T2 this terminal vertex is moved from Pb -branch to the Pa -branch. Note that:
• T1 [{v} ∪ V (Pa ) ∪ V (Pb )] = Pa+b+1 = T2 [{v} ∪ V (Pa ) ∪ V (Pb )]. • distT1 (x, y) = distT2 (x, y), if x ∈ {v} ∪ V (Pa ) and y ∈ / {v} ∪ V (Pa ) ∪ V (Pb ). • distT1 (x, y) < distT2 (x, y), if x ∈ V (Pb ) and y ∈ / {v} ∪ V (Pa ) ∪ V (Pb ). This implies that the transformation T1 → T2 increases the hyper-Wiener index. Theorem 3.5. Let G be a connected graph with order n and △(G) ≥ d ≥ 2. Then WW (G) ≤ Φ (n, d), with equality if and only if G = Tn,d . Proof. By Lemma 2.3, Φ (n, d + 1) ≤ Φ (n, d). Without loss of generality, we may assume that △(G) = d, and it suffices to show that WW (G) ≤ Φ (n, d) and the equality holds if and only if G = Tn,d . Let T be a spanning tree with maximum degree d in G, then WW (G) ≤ WW (T ). Taking a vertex v with neighbors x11 , x21 , . . . , xd1 in T , we denote Ti be the component of T − v containing xi1 with order ni for each i = 1, 2, . . . , d. By replacing each component Ti with a path Pni = xi1 xi2 · · · xini with the same pendent vertex xi1 , we obtain a new tree T3 . By Transformation I, we know that WW (T ) < WW (T3 ). Without loss of generality, we assume that T3 ̸= Tn,d , otherwise we are done. For sake of simplicity, we assume that n1 ≥ n2 ≥ · · · ≥ nd . Let T4 = T3 − x21 x22 + x22 x1n1 . By Transformation II, we have that WW (T3 ) < WW (T4 ). Whenever the resulting new tree T4 ̸= Tn,d , we repeat Transformation II again, which increases the hyper-Wiener index strictly, until we will obtain the tree Tn,d . This completes the proof.
G. Su et al. / Theoretical Computer Science 471 (2013) 74–83
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Before closing this section, we prove the following: Theorem 3.6. Let G1 and G2 be two connected edge-disjoint graphs with the same order n. If △(G1 ) + △(G2 ) ≥ c for some positive real number c , then WW (G1 ) + WW (G2 ) < 3n2 + 21 σ ( 2c , 2c ). Proof. Let d1 = △(G1 ) and d2 = △(G2 ). By Lemma 2.4 and Theorem 3.5, we have WW (G1 ) + WW (G2 ) ≤ Φ (n, d1 ) + Φ (n, d2 ) 1 1 = [Φ1 (n, d1 ) + Φ2 (n, d1 )] + [Φ1 (n, d2 ) + Φ2 (n, d2 )] 2 2 1 2 2 2 < [d1 + d2 + f (d1 , d2 ) + 2d1 + 2d22 + g (d1 , d2 )] 2 1 = [3d21 + 3d22 + σ (d1 , d2 )] 2 1 c c < 3n2 + σ . , 2 2 2 This completes the proof of Theorem 3.6. 4. The main result In this section, we present our main result. Theorem 4.1. Let (G1 , G2 , G3 ) be a 3-decomposition of Kn such that each cell Gi is connected. Then for any n ≥ 70, we have
7
n
2
n+2 n ≤ WW (G1 ) + WW (G2 ) + WW (G3 ) ≤ 2 + + 4(n − 1), 4
2
with right equality if and only G1 = G2 = Pn , and with left equality if and only diam(G1 ) = diam(G2 ) = 2. To complete the proof, we need the following two auxiliary results. Let Gn,d denote the graph with order n and diameter d ≥ 2, and ei (G) the number of pairs of vertices whose distance is i for 1 ≤ i ≤ d in G, thus e1 (G) is the number of edges of G. Theorem 4.2. WW (Gn,d ) ≥ WW (Gn,2 ) holds for d ≥ 2. Proof. By the definition of hyper-Wiener index, we have WW (Gn,d ) =
1
W (Gn,d ) +
2 1
1 2
W2 (Gn,d )
− e(G) − e2 (G) − · · · − ed−1 (G) 2 2 1 n + e(G) + 22 e2 (G) + · · · + (d − 1)2 ed−1 (G) + d2 − e(G) − e2 (G) − · · · − ed−1 (G) 2 2 d2 + d − 6 d2 + d − 3 n d2 + d − 3 n = 3 − 2e(G) + − e(G) − e2 (G) − · · · − ded−1 (G) 2 2 2 2 2 n n d2 + d − 3 ≥ 3 − 2e(G) + − e(G) − e2 (G) − e3 (G) − · · · − ed−1 (G) 2 2 2 n n ≥ 3 − 2e(G) By the fact − e(G) − e2 (G) − e3 (G) − · · · − ed−1 (G) > 0 . =
e(G) + 2e2 (G) + · · · + (d − 1)ed−1 (G) + d
2
n
2
On the other hand WW (Gn,2 ) =
1 2
e(G) + 2
n
2
1 n n 2 − e(G) + e(G) + 2 − e(G) = 3 − 2e(G). 2
2
2
This completes the proof of Theorem 4.2. Theorem 4.3. Let (G1 , G2 , G3 ) be a 3-decomposition of Kn such that each cell Gi is connected. Then △(Gi )+△(Gj ) ≥ n−1−δ(Gk ).
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G. Su et al. / Theoretical Computer Science 471 (2013) 74–83
Proof. Without loss of generality, let u be a vertex in Gi with maximum degree. We distinguish the following two cases: Case 1. △(Gj ) = degGj (u), then △(Gi ) + △(Gj ) + δ(Gk ) = degGi (u) + degGj (u) + degGk (u) = n − 1. Case 2. △(Gj ) ̸= degGj (u), then there exists a vertex v distinct with u in Gj such that △(Gj ) = degGj (v). If δ(Gk ) = degGk (v), we have
△(Gi ) + △(Gj ) + δ(Gk ) = degGi (u) + degGj (v) + degGk (v) ≥ degGi (v) + degGj (v) + degGk (v) = n − 1. If δ(Gk ) ̸= degGk (v), then there exists a vertex w, distinct with u and v, in Gk with minimum degree. Thus
△(Gi ) + △(Gj ) + δ(Gk ) = degGi (u) + degGj (v) + degGk (w) ≥ degGi (w) + degGj (w) + degGk (w) = n − 1. This completes the proof of Theorem 4.3.
Proof of Theorem 4.1. Let us firstly prove the upper bound. For sake of simplicity, let
P (n) = 2
n+2 4
+
n
2
+ 4(n − 1) =
n4 + 2n3 + 5n2 + 40n − 48 12
.
We distinguish the following three cases: Case 1. There exist at least two cells of {G1 , G2 , G3 }, say G1 , G2 , such that δ(Gi ) ≥ 11 for i ∈ {1, 2}. By Lemma 2.2, diam(Gi ) ≤ δ(G3n)+1 ≤ i WW (Gi ) =
1 2
W (Gi ) +
1 2
n 4
W2 (Gi ) ≤
, and then
1
·
2
n
2
·
n 4
+
1 2
·
n
2
·
n2
=
16
1 64
n4 +
3 64
n3 −
1 16
n2 .
Hence, for any n ≥ △(G) + 1 ≥ 12, we have WW (G1 ) + WW (G2 ) + WW (G3 ) ≤
7 96
n4 +
17 96
n3 −
1 6
n2 −
1 12
n
36 we have WW (G1 ) + WW (G2 ) + WW (G3 )
≤ WW (Tn,△(G1 ) ) + WW (Tn,△(G2 ) ) + WW (Tn,△(G3 ) ) ≤ 2WW (Tn,⌊ 2n −6⌋ ) + WW (Tn,10 ) 5 4 7 3 69 2 20 1148 1 4 4 383 2 1233 △(G3 ). By Theorem 3.5, for n ≥ 70 we have WW (G1 ) + WW (G2 ) + WW (G3 ) ≤ WW (Tn,△(G1 ) ) + WW (Tn,△(G2 ) ) + WW (Pn )
≤ 2WW (Tn,⌊ 2n −6⌋ ) + WW (Pn ) 5 4 7 3 69 2 20 1148 1 4 1 3 1 2 1 + =2 n + n + n − n− n + n − n − n 384
12
8
5
3
24
12
24
12
1 13 4 5 413 2 161 · n + n3 + n − n 12 16 4 24 12 n4 + 2n3 + 5n2 + 40n − 48 < = P (n). 12
=
Subcase 3.4. △(G1 ) ≥ 2n − 6 > △(G2 ). It is obvious that △(G2 ) + △(G3 ) < n − 12. By Theorem 4.3, we have △(G2 ) + △(G3 ) ≥ n − 1 − δ(G1 ) > n − 12. Thus △(G2 ) + △(G3 ) = n − 12, hence △(G1 ) ≥ 2n − 6 > △(G2 ) ≥ 2n − 6. By the same arguments as subcase 3.1 and 3.2, the proof can be obtained. Finally, we consider the lower bound. By Theorem 4.2, we have
WW (G1 ) + WW (G2 ) + WW (G3 ) ≥ 9
n
2
− 2[e(G1 ) + e(G2 ) + e(G3 )] = 7
n
2
.
This completes the proof of Theorem 4.1 as desired. 5. Conclusions and open problem The key contribution of this paper is the following. We introduced the concept of k-decomposition for a graph, and then presented the Nordhaus–Gaddum-type inequality of a 3-decomposition of Kn for the hyper-Wiener index. However, exploring the corresponding Nordhaus–Gaddum-type inequality of a k-decomposition is still an open problem for k ≥ 4. We leave those questions for future research. Conjecture 5.1. Let (G1 , G2 , . . . , Gk ) be a k-decomposition of Kn such that each cell Gi is connected. Then for any sufficiently large n with respect to k, we have
n n+2 n (3k − 2) ≤ WW (G1 ) + WW (G2 ) + · · · + WW (Gk ) ≤ (k − 1) + + 2(k − 1)(n − 1). 2
The lower and upper bounds are sharp.
4
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G. Su et al. / Theoretical Computer Science 471 (2013) 74–83
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If Conjecture 5.1 holds, we can extend Theorem A for arbitrarily large k. The upper bound is attained when G1 = G2 =
· · · = Gk−1 = Pn , since if 2n − (k − 1)(n − 1) ≥ 21 nδ(Gk ) ≥ any sufficiently large n. Hence
n2 4
, i.e., k ≤ 4n , then by Lemma 2.1 we have diam(Gk ) = 2 for
WW (G1 ) + WW (G2 ) + · · · + WW (Gk ) = WW (G1 ) + WW (G2 ) + · · · + WW (Gk−1 ) + 3
n
2
− 2e(Gk )
n = [WW (G1 ) + 2e(G1 )] + [WW (G2 ) + 2e(G2 )] + · · · + [WW (Gk−1 ) + 2e(Gk−1 )] + 2 n = (k − 1)WW (Pn ) + 2(k − 1)(n − 1) + . 2
The lower bound is trivial and is attained n by any k-decomposition n(G1 , G2 , . . . , Gk ) of Kn with diam(Gi ) = 2 for k i = 1, 2, . . . , k, this is because WW (Gi ) = 3 2 − 2e(Gi ) and i=1 e(Gi ) 2 . Acknowledgments The authors thank the referees for their careful reading and valuable suggestions. This work was supported by Natural Science Fund of China (No. 11071016, 11171129) and Xinjiang Department of Education Scientific Research Projects in Universities (No. XJEDU2012I38). References [1] Z. An, B. Wu, D. Bin, Y. Wang, G. Su, Nordhaus–Gaddum-type theorem for diameter of graphs when decomposing into many parts, Discrete Math. Algorithms Appl. 3 (2011) 305–310. [2] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, American Elsevier [M], New York, 1976. [3] A.A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory and applications, Acta. Appl. Math. 66 (2001) 211–249. [4] P. Erdös, J. Pach, R. Pollack, Z. Tuza, Radius, diameter and minimum degree, J. Combin. Theory Ser. B 47 (1989) 73–79. [5] I. Gutman, W. Linert, I. Lukovits, A.A. Dobrynin, Trees with extremal hyper-Wiener index: mathematical basis and chemical, J. Chem. Inf. Comput. Sci. 37 (1997) 349–354. [6] M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, The hyper-Wiener index of graph operations, Comput. Math. Appl. 56 (2008) 1402–1407. [7] S. Klavžar, P. Zigert, I. Gutman, An algorithm for the calculation of the hyper-Wiener index of benzenoid hydrocarbons, Comput. Chem. 24 (2000) 229–233. [8] D.J. Klein, I. Lukovits, I. Gutman, On the definition of the hyper-Wiener index for cycle-containing structures, J. Chem. Inf. Comput. Sci. 35 (1995) 50–52. [9] X. Li, A.F. Jalbout, Bond order weighted hyper-Wiener index, J. Mol. Structure (Theochem) 634 (2003) 121–125. [10] D. Li, B. Wu, X. Yang, X. An, Nordhaus–Gaddum-type theorem for Wiener index of graphs when decomposing into three parts, Discrete Appl. Math. 159 (2011) 1594–1600. [11] W. Linert, I. Lukovits, Formulas for the hyper-Wiener and hyper-Detour indices of fused bicyclic structures, MATCH Comm. Math. Comput. Chem. 35 (1997) 65–74. [12] M. Liu, X. Tan, The first to (k + 1)-th smallest Wiener (hyper-Wiener) indices of connected graphs, Kragujevac J. Math. 32 (2009) 109–115. [13] D.E. Needham, I.C. Wei, P.G. Seybold, Molecular modeling of the physical properties of the alkanes, J. Amer. Chem. Soc. 110 (1988) 4186–4194. [14] E.A. Nordhaus, J.W. Gaddum, On complementary graphs, Amer. Math. Monthly 63 (1956) 175–177. [15] M. Randić, Novel molecular descriptor for structure-property studies, Chem. Phys. Lett. 211 (1993) 478–483. [16] G. Rücker, C. Rücker, On the topological indices, boiling points, and cycloalkanes, J. Chem. Inf. Comput. Sci. 39 (1999) 788–802. [17] G. Strang, Calculus, Wellesley-Cambridge Press, 1991. [18] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 (1947) 17–20. [19] W. Zhang, B. Wu, X. An, The hyper-Wiener index of the kth power of a graph, Discrete Math. Algor. Appl. 3 (2011) 17–23.
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Nordhaus–Gaddum-type inequality for the hyper-Wiener index of graphs when decomposing into three parts Guifu Su a,∗ , Liming Xiong a,b , Yi Sun c , Daobin Li d a
School of Mathematics, Beijing Institute of Technology, Beijing, 100081, PR China
b
Department of Mathematics, Jiangxi University, Jiangxi, 330026, PR China
c
College of Mathematics and System Science, Xinjiang University, Xinjiang, 830046, PR China
d
School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing, 100083, PR China
article
abstract
info
Article history: Received 1 May 2012 Received in revised form 25 September 2012 Accepted 11 October 2012 Communicated by D.-Z. Du Keywords: Wiener index Hyper-Wiener index k-decomposition Nordhaus–Gaddum-type inequality
Let k ≥ 2 be an integer, a k-decomposition (G1 , G2 , · · · , Gk ) of a graph G is a partition of its edge set to form k spanning subgraphs G1 , G2 , . . . , Gk . The hyper-Wiener index WW is one of the recently conceived distance-based graph invariants (Randi 1993 [15]): WW = WW (G) := 12 W (G) + 12 W2 (G), where W is the Wiener index (Wiener 1947 [18]) and W2 is the sum of squares of distance of all pairs of vertices in G. In this paper, we investigate the Nordhaus–Gaddum-type inequality of a 3-decomposition of Kn for the hyper-Wiener index: 7
n 2
≤ WW (G1 ) + WW (G2 ) + WW (G3 ) ≤ 2
n+2 4
+
n 2
+ 4(n − 1).
The corresponding extremal graphs are characterized. Published by Elsevier B.V.
1. Introduction Throughout the paper all graphs considered are finite and simple. Let G = (V (G), E (G)) be a graph, we use |V | = |V (G)| and e(G) = |E (G)| to denote its order and size, respectively. The degree of a vertex u is the number of edges incident with it in G, denoted by degG (u), or deg(u) when no confusion is possible. As usual, △(G) and δ(G) denote the maximum and the minimum degree of G, respectively. The distance distG (u, v) between vertices u and v is the length of the shortest path in G connecting them. The diameter diam(G) of a graph G is the maximal distance between any two vertices. The complement of G, denoted by G, is a simple graph on the same set of vertices V in which two vertices are adjacent if and only if they are not adjacent in G. Let k ≥ 2 be an integer, a k-decomposition (G1 , G2 , . . . , Gk ) of a graph G is a partition of its edge set to form k spanning subgraphs G1 , G2 , . . . , Gk , and each Gi is said to be a cell. In other words, each Gi has the same vertices as G, and each edge of G belongs to exactly one of G1 , G2 , . . . , Gk . In particular, (G1 , G2 ) is a 2-decomposition of Kn if and only if G1 is the complement of G2 . Other terminology and notations needed will be introduced as it naturally occurs in the following and we use [2] for those not defined here. The Wiener index W of a connected graph G is the sum of distance of all pairs of vertices in it [18]: W = W (G) :=
distG (u, v),
{u,v}∈V (G)
∗ Correspondence to: Computational Systems Biology Lab, Department of Biochemistry and Molecular Biology, The University of Georgia, GA, USA. Tel.: +86 01013717961438. E-mail address: [email protected] (G. Su). 0304-3975/$ – see front matter. Published by Elsevier B.V. doi:10.1016/j.tcs.2012.10.049
G. Su et al. / Theoretical Computer Science 471 (2013) 74–83
75
which was introduced by the chemist Wiener in 1947. Its chemical applications and mathematical properties were raised in [3,13,16]. The hyper-Wiener index WW is one of the recently conceived distance-based graph invariants [15]: WW = WW (G) :=
1 2
W (G) +
1 2
W2 (G),
2 where W2 (G) = {u,v}∈V (G) distG (u, v). This index was introduced by Randić and used as a structure-descriptor for predicting physico-chemical properties of organic compounds, often those significant for pharmacology, agriculture, environmental protection and so on. It rapidly gained popularity and numerous results on it were stated in [6–9,11]. Let l be a positive integer not less than 1, and β be a parameter of graph G, to determine the extremal (maximum or minimum) value of
l
β(Gi ) : (G1 , G2 , . . . , Gl ) is a l-decomposition of G
i=1
is a fundamental problem in graph theory. The particular case when G = Kn attracts much attention on various graph parameters. It was Nordhaus and Gaddum [14] who first initiated such kind of research on chromatic number of graphs for the case when l = 2 and G = Kn . They proved that:
√
2 n ≤ χ (G) + χ(G) ≤ n + 1
and
n ≤ χ (G) · χ (G) ≤
n+1 2
2
,
where χ denotes the chromatic number of graph G. Zhang and her co-workers [19] showed that: Theorem A (Zhang et al. [19]). Let (G1 , G2 ) be a 2-decomposition of Kn such that each cell Gi is connected. Then for any sufficiently large n, we have
4
n
2
n+4 . ≤ WW (G1 ) + WW (G2 ) ≤ 2 4
The lower and the upper bounds are sharp. Later, Li et al. paid their attention to the following for the diameter in [10]. They said: Theorem B (Li et al. [10]). Let (G1 , G2 , G3 ) be a 3-decomposition of Kn such that each cell Gi is connected. Then for any sufficiently large n, we have
5
n
2
≤ W (G1 ) + W (G2 ) + W (G3 ) ≤
n3 − n 3
+
n
2
+ 2(n − 1).
The lower and the upper bounds are sharp. Motivated by Theorems A and B, in this paper, we consider the Nordhaus–Gaddum-type inequality of a 3-decomposition of Kn for the hyper-Wiener index. The corresponding extremal graphs are characterized. 2. Preliminary lemmas In this section we list or prove some lemmas as basic but necessary preliminaries, which will be used in the subsequent proofs. Lemma 2.1 (An et al. [1]). Let G be a simple graph with order n. If δ(G) ≥
n 2
, then diam(G) ≤ 2.
− 1. Lemma 2.2 (Erdös et al. [4]). Let G be a connected graph with order n. Then diam(G) ≤ δ(G3n )+1 G
:G
Let us introduce some useful notations. A splice a1 :a 2 of two connected graphs G1 and G2 is a graph obtained by 1 2 identifying the vertices a1 ∈ V1 and a2 ∈ V2 . K :P Let n and d be two integers with n > d, we denote Tn,d = 1b,d :bn−d the graph obtained from identifying a leaf b1 of the 1 2 star K1,d with a leaf b2 of the path Pn−d . It is trivial to see that the graph Tn,d has maximum degree d and order n. In particular, Tn,d = Pn when d = 2. By some calculations, we have W (Tn,d ) = (d − 1)(d − 2) +
(d − 1)(n − d + 1)(n − d + 2) 2
+
(n − d + 1)3 − (n − d + 1) 6
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G. Su et al. / Theoretical Computer Science 471 (2013) 74–83
and W2 (Tn,d ) = 2(d − 1)(d − 2) +
(d − 1)(n − d + 1)(n − d + 2)(2n − 2d + 3) 6
+
(n − d)(n − d + 2)(n − d + 1)2 12
.
For simplicity, we denote Φ1 (n, d) = W (Tn,d ) and Φ2 (n, d) = W2 (Tn,d ) in sequel. Then we have the following consequence. Lemma 2.3. For 2 ≤ d ≤ n − 1, Φ (n, d) =
1 Φ 2 1
(n, d) + 21 Φ2 (n, d) is a decreasing function on d.
Proof. This result can be verified directly, so we omit the proof here. Lemma 2.4. Let n be a fixed positive integer and c a real number with n > 2c > 0. Then the function
σ (x 1 , x 2 ) =
2 xi (n − xi + 2)2 i =1
+
2
takes the maximum value at x1 = x2 =
c 2
(n − xi + 2)3
+
6
xi (n − xi + 2)3 3
+
(n − xi + 2)4
12
if x1 + x2 ≥ c .
Proof. For simplicity, let f (x1 , x2 ) =
2 xi (n − xi + 2)2 i=1
2
+
(n − xi + 2)3
+
(n − xi + 2)4
6
and g (x1 , x2 ) =
2 xi (n − xi + 2)3 i=1
3
12
.
Li et al. [10] proved that the function f (x1 , x2 ) takes the maximum value at (x1 , x2 ) = ( 2c , 2c ). Here we only need to consider the problem: maximize g (x1 , x2 ) subject to x1 + x2 = c . By means of Lagrange multiplier [17], we show that the maximum value of g (x1 , x2 ) is also attained at (x1 , x2 ) = ( 2c , 2c ). Put R(x1 , x2 , λ) = g (x1 , x2 ) + λ(x1 + x2 − c ). Set the derivative dR = 0, which yields the following system of equations:
∂R = −x1 (n − x1 + 2)2 + λ = 0. ∂ x1 ∂R = −x2 (n − x2 + 2)2 + λ = 0. ∂ x2 ∂R = x1 + x2 − c = 0. ∂λ Combining the above equations, we have x1 = x2 =
c 2
by easy calculations.
Note that the function g (x1 , x2 ) takes the maximum value when x1 = x2 = where A=
c 2
, since △ = B2 − AC < 0 and A < 0 [17],
∂ 2R 3 = −(n + 2 − 3x1 )(n + 2 − x1 ) = − n + 2 − c (n + 2 − c ) < 0. 2 2 ∂ x1
∂ 2R ∂ 2G = = 0. ∂ x1 x2 ∂ x2 x1 ∂ 2R 3 C = = −(n + 2 − 3x2 )(n + 2 − x2 ) = − n + 2 − c (n + 2 − c ) < 0. 2 2 ∂ x2 B=
Hence, σ (x1 , x2 ) = f (x1 , x2 ) + g (x1 , x2 ) takes the maximum value at x1 = x2 = This completes the proof of Lemma 2.4.
c 2
subject to x1 + x2 ≥ c .
Lemma 2.5 (An et al. [1]). Let Kn be the complete graph with order n and k ≥ 2 any fixed integer. Then for any sufficiently large n with respect to k, there exists a k-decomposition (G1 , G2 , . . . , Gk ) of Kn such that diam(Gi ) = 2 for each i = 1, 2, . . . , k. Lemma 2.6. Let G be a connected non-complete graph and e be its non-cut-edge. Then WW (G − e) + 2e(G − e) ≥ WW (G) + 2e(G).
G. Su et al. / Theoretical Computer Science 471 (2013) 74–83
77
Proof. Let e = uv ∈ E (G), by some computation W (G) =
distG (u, x) +
x̸=v
distG (v, y) + distG (u, v) + distG (v, u) +
distG (p, q)
p̸=u,q̸=v
y̸=u
and W (G − e) =
distG−e (u, x) +
x̸=v
distG−e (v, y) + distG−e (u, v) + distG−e (v, u) +
distG−e (p, q),
p̸=u,q̸=v
y̸=u
which implies that W (G − e) > W (G) + 1. By analogous argument we get W2 (G − e) > W2 (G) + 1. Then W (G − e) + 2e(G) > W (G) + 2e(G) + 1
⇒ W (G − e) + 2e(G) − 1 ≥ W (G) + 2e(G) + 1 ⇒ W (G − e) + 2e(G − e) ≥ W (G) + 2e(G). In a similar way, we have W2 (G − e) + 2e(G − e) ≥ W2 (G) + 2e(G). Therefore WW (G − e) + 2e(G − e) =
= ≥
1 2 1 2 1 2 1
W (G − e) +
1 2
W2 (G − e) + 2e(G − e) 1
[W (G − e) + 2e(G − e)] + [W2 (G − e) + 2e(G − e)] 2
1
[W (G) + 2e(G)] + [W2 (G) + 2e(G)] 2
1 W (G) + W2 (G) + 2e(G) 2 2 = WW (G) + 2e(G).
=
This completes the proof of Lemma 2.6. Lemma 2.7. Let (G1 , G2 , G3 ) be a 3-decomposition of Kn such that each cell Gi is connected. Then there exists at most one cell Gi with δ(Gi ) ≥ 2n . Proof. By contradiction, assume there are at least two cells of {G1 , G2 , G3 }, say G1 and G2 , such that δ(Gi ) ≥ By the Handshaking Lemma, we have 2e(G1 ) + 2e(G2 ) =
u∈V (G1 )
degG1 (u) +
u∈V (G2 )
degG2 (u) ≥
1 2
n2 +
1 2
n 2
for i = 1, 2.
n2 = n2 > n(n − 1) = 2e(Kn ),
which is a contradiction. This completes the proof of Lemma 2.7. 3. Upper bounds on hyper-Wiener index Let Sn denote the star with n vertices. Gutman and his co-workers determined trees with minimal and maximal hyperWiener indices in [5]. Theorem 3.1 (Gutman et al. [5]). Let T be a tree with order n. Then WW (Sn ) ≤ WW (T ) ≤ WW (Pn ). Theorem 3.2 (Liu et al. [12]). Let G′ be a connected spanning subgraph of G. Then WW (G) ≤ WW (G′ ). Combining the above theorems, we get: Corollary 3.3. Let G be a connected graph with order n. Then WW (G) ≤
, with equality if and only if G = Pn . 4 n+2 Proof. Note that WW (Pn ) = 4 . Let T be a spanning tree of G, then WW (G) ≤ WW (T ) by Theorem 3.2. By Theorem 3.1, we have WW (G) ≤ WW (Pn ), as desired. n+2
Let α ≥ 1 be an integer, the α -transmission of a vertex u, denoted by Trα (u : G), is the sum of distances from it to all the other vertices in graph G, i.e., Trα (u) = Trα (u : G) =
distαG (u, v).
v∈V
Theorem 3.4. Let Ti be a tree with order ni and a specified vertex xi for i = 1, 2. Then WW ( and xi is a leaf of Ti .
T1 :T2 x1 :x2
) is maximized only if Ti = Pni
78
G. Su et al. / Theoretical Computer Science 471 (2013) 74–83
Fig. 1. The transformation T1 → T2 increases the WW -value.
Proof. For convenience, let V (T1 ) = {y1 , y2 , . . . , yi0 = x1 , . . . , yn1 }, V (T2 ) = {z1 , z2 , . . . , zj0 = x2 , · · · , zn2 } and G=
T1 :T2 . It is easily seen that x1 :x2
W (G) = W (T1 ) + W (T2 ) + (n2 − 1)Tr(x1 : T1 ) + (n1 − 1)Tr(x2 : T2 ). On the other hand, dist2G (yi , zj ) = [distT1 (x1 , yi ) + distT2 (x1 , zj )]2 = dist2T1 (x1 , yi ) + dist2T2 (x1 , zj ) + 2distT1 (x1 , yi ) · distT2 (x1 , zj ). Hence, the contribution from pairs of vertices yi and zj to W2 (G) equals to the sum: n1 n2
dist2G (yi , zj ) =
i=1 j=1
n1 n2 [distT1 (x1 , yi ) + distT2 (x1 , zj )]2 i=1 j=1
=
n1 n2 i=1 j=1
dist2T1 (x1 , yi ) +
n1 n2
dist2T2 (x1 , zj ) + 2
i=1 j=1
n1 n2
distT1 (x1 , yi ) · distT2 (x1 , zj )
i=1 j=1
= n2 Tr2 (x1 : T1 ) + n1 Tr2 (x2 : T2 ) + 2Tr(x1 : T1 ) · Tr(x2 : T2 ). Thus, we have W2 (G) = W2 (T1 ) + W2 (T2 ) + (n2 − 1)Tr2 (x1 : T1 ) + (n1 − 1)Tr2 (x2 : T2 ) + 2Tr(x1 : T1 ) · Tr(x2 : T2 ). Hence, WW (G) = WW (T1 ) + WW (T2 ) + Tr(x1 : T1 ) · Tr(x2 : T2 ) 1 1 + (n2 − 1)[Tr2 (x1 : T1 ) + Tr(x1 : T1 )] + (n1 − 1)[Tr2 (x2 : T2 ) + Tr(x2 : T2 )]. 2 2 By Corollary 3.3, we complete the proof as desired. Next, we will introduce two auxiliary transformations, which increase the WW -value of graphs. Transformation I. Let T be a tree and v a vertex with maximum degree in T . Let T0 be a component of T − v. By replacing T0 with a path with the same order and the same pendent vertex as T0 under the definition of splice, we obtain a new tree ′ ′ T . By Theorem 3.4, we confirm that the transformation T → T increases the hyper-Wiener index. Transformation II. Consider the trees T1 and T2 depicted in Fig. 1. These two trees differ only in the position of a terminal vertex: in T2 this terminal vertex is moved from Pb -branch to the Pa -branch. Note that:
• T1 [{v} ∪ V (Pa ) ∪ V (Pb )] = Pa+b+1 = T2 [{v} ∪ V (Pa ) ∪ V (Pb )]. • distT1 (x, y) = distT2 (x, y), if x ∈ {v} ∪ V (Pa ) and y ∈ / {v} ∪ V (Pa ) ∪ V (Pb ). • distT1 (x, y) < distT2 (x, y), if x ∈ V (Pb ) and y ∈ / {v} ∪ V (Pa ) ∪ V (Pb ). This implies that the transformation T1 → T2 increases the hyper-Wiener index. Theorem 3.5. Let G be a connected graph with order n and △(G) ≥ d ≥ 2. Then WW (G) ≤ Φ (n, d), with equality if and only if G = Tn,d . Proof. By Lemma 2.3, Φ (n, d + 1) ≤ Φ (n, d). Without loss of generality, we may assume that △(G) = d, and it suffices to show that WW (G) ≤ Φ (n, d) and the equality holds if and only if G = Tn,d . Let T be a spanning tree with maximum degree d in G, then WW (G) ≤ WW (T ). Taking a vertex v with neighbors x11 , x21 , . . . , xd1 in T , we denote Ti be the component of T − v containing xi1 with order ni for each i = 1, 2, . . . , d. By replacing each component Ti with a path Pni = xi1 xi2 · · · xini with the same pendent vertex xi1 , we obtain a new tree T3 . By Transformation I, we know that WW (T ) < WW (T3 ). Without loss of generality, we assume that T3 ̸= Tn,d , otherwise we are done. For sake of simplicity, we assume that n1 ≥ n2 ≥ · · · ≥ nd . Let T4 = T3 − x21 x22 + x22 x1n1 . By Transformation II, we have that WW (T3 ) < WW (T4 ). Whenever the resulting new tree T4 ̸= Tn,d , we repeat Transformation II again, which increases the hyper-Wiener index strictly, until we will obtain the tree Tn,d . This completes the proof.
G. Su et al. / Theoretical Computer Science 471 (2013) 74–83
79
Before closing this section, we prove the following: Theorem 3.6. Let G1 and G2 be two connected edge-disjoint graphs with the same order n. If △(G1 ) + △(G2 ) ≥ c for some positive real number c , then WW (G1 ) + WW (G2 ) < 3n2 + 21 σ ( 2c , 2c ). Proof. Let d1 = △(G1 ) and d2 = △(G2 ). By Lemma 2.4 and Theorem 3.5, we have WW (G1 ) + WW (G2 ) ≤ Φ (n, d1 ) + Φ (n, d2 ) 1 1 = [Φ1 (n, d1 ) + Φ2 (n, d1 )] + [Φ1 (n, d2 ) + Φ2 (n, d2 )] 2 2 1 2 2 2 < [d1 + d2 + f (d1 , d2 ) + 2d1 + 2d22 + g (d1 , d2 )] 2 1 = [3d21 + 3d22 + σ (d1 , d2 )] 2 1 c c < 3n2 + σ . , 2 2 2 This completes the proof of Theorem 3.6. 4. The main result In this section, we present our main result. Theorem 4.1. Let (G1 , G2 , G3 ) be a 3-decomposition of Kn such that each cell Gi is connected. Then for any n ≥ 70, we have
7
n
2
n+2 n ≤ WW (G1 ) + WW (G2 ) + WW (G3 ) ≤ 2 + + 4(n − 1), 4
2
with right equality if and only G1 = G2 = Pn , and with left equality if and only diam(G1 ) = diam(G2 ) = 2. To complete the proof, we need the following two auxiliary results. Let Gn,d denote the graph with order n and diameter d ≥ 2, and ei (G) the number of pairs of vertices whose distance is i for 1 ≤ i ≤ d in G, thus e1 (G) is the number of edges of G. Theorem 4.2. WW (Gn,d ) ≥ WW (Gn,2 ) holds for d ≥ 2. Proof. By the definition of hyper-Wiener index, we have WW (Gn,d ) =
1
W (Gn,d ) +
2 1
1 2
W2 (Gn,d )
− e(G) − e2 (G) − · · · − ed−1 (G) 2 2 1 n + e(G) + 22 e2 (G) + · · · + (d − 1)2 ed−1 (G) + d2 − e(G) − e2 (G) − · · · − ed−1 (G) 2 2 d2 + d − 6 d2 + d − 3 n d2 + d − 3 n = 3 − 2e(G) + − e(G) − e2 (G) − · · · − ded−1 (G) 2 2 2 2 2 n n d2 + d − 3 ≥ 3 − 2e(G) + − e(G) − e2 (G) − e3 (G) − · · · − ed−1 (G) 2 2 2 n n ≥ 3 − 2e(G) By the fact − e(G) − e2 (G) − e3 (G) − · · · − ed−1 (G) > 0 . =
e(G) + 2e2 (G) + · · · + (d − 1)ed−1 (G) + d
2
n
2
On the other hand WW (Gn,2 ) =
1 2
e(G) + 2
n
2
1 n n 2 − e(G) + e(G) + 2 − e(G) = 3 − 2e(G). 2
2
2
This completes the proof of Theorem 4.2. Theorem 4.3. Let (G1 , G2 , G3 ) be a 3-decomposition of Kn such that each cell Gi is connected. Then △(Gi )+△(Gj ) ≥ n−1−δ(Gk ).
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G. Su et al. / Theoretical Computer Science 471 (2013) 74–83
Proof. Without loss of generality, let u be a vertex in Gi with maximum degree. We distinguish the following two cases: Case 1. △(Gj ) = degGj (u), then △(Gi ) + △(Gj ) + δ(Gk ) = degGi (u) + degGj (u) + degGk (u) = n − 1. Case 2. △(Gj ) ̸= degGj (u), then there exists a vertex v distinct with u in Gj such that △(Gj ) = degGj (v). If δ(Gk ) = degGk (v), we have
△(Gi ) + △(Gj ) + δ(Gk ) = degGi (u) + degGj (v) + degGk (v) ≥ degGi (v) + degGj (v) + degGk (v) = n − 1. If δ(Gk ) ̸= degGk (v), then there exists a vertex w, distinct with u and v, in Gk with minimum degree. Thus
△(Gi ) + △(Gj ) + δ(Gk ) = degGi (u) + degGj (v) + degGk (w) ≥ degGi (w) + degGj (w) + degGk (w) = n − 1. This completes the proof of Theorem 4.3.
Proof of Theorem 4.1. Let us firstly prove the upper bound. For sake of simplicity, let
P (n) = 2
n+2 4
+
n
2
+ 4(n − 1) =
n4 + 2n3 + 5n2 + 40n − 48 12
.
We distinguish the following three cases: Case 1. There exist at least two cells of {G1 , G2 , G3 }, say G1 , G2 , such that δ(Gi ) ≥ 11 for i ∈ {1, 2}. By Lemma 2.2, diam(Gi ) ≤ δ(G3n)+1 ≤ i WW (Gi ) =
1 2
W (Gi ) +
1 2
n 4
W2 (Gi ) ≤
, and then
1
·
2
n
2
·
n 4
+
1 2
·
n
2
·
n2
=
16
1 64
n4 +
3 64
n3 −
1 16
n2 .
Hence, for any n ≥ △(G) + 1 ≥ 12, we have WW (G1 ) + WW (G2 ) + WW (G3 ) ≤
7 96
n4 +
17 96
n3 −
1 6
n2 −
1 12
n
36 we have WW (G1 ) + WW (G2 ) + WW (G3 )
≤ WW (Tn,△(G1 ) ) + WW (Tn,△(G2 ) ) + WW (Tn,△(G3 ) ) ≤ 2WW (Tn,⌊ 2n −6⌋ ) + WW (Tn,10 ) 5 4 7 3 69 2 20 1148 1 4 4 383 2 1233 △(G3 ). By Theorem 3.5, for n ≥ 70 we have WW (G1 ) + WW (G2 ) + WW (G3 ) ≤ WW (Tn,△(G1 ) ) + WW (Tn,△(G2 ) ) + WW (Pn )
≤ 2WW (Tn,⌊ 2n −6⌋ ) + WW (Pn ) 5 4 7 3 69 2 20 1148 1 4 1 3 1 2 1 + =2 n + n + n − n− n + n − n − n 384
12
8
5
3
24
12
24
12
1 13 4 5 413 2 161 · n + n3 + n − n 12 16 4 24 12 n4 + 2n3 + 5n2 + 40n − 48 < = P (n). 12
=
Subcase 3.4. △(G1 ) ≥ 2n − 6 > △(G2 ). It is obvious that △(G2 ) + △(G3 ) < n − 12. By Theorem 4.3, we have △(G2 ) + △(G3 ) ≥ n − 1 − δ(G1 ) > n − 12. Thus △(G2 ) + △(G3 ) = n − 12, hence △(G1 ) ≥ 2n − 6 > △(G2 ) ≥ 2n − 6. By the same arguments as subcase 3.1 and 3.2, the proof can be obtained. Finally, we consider the lower bound. By Theorem 4.2, we have
WW (G1 ) + WW (G2 ) + WW (G3 ) ≥ 9
n
2
− 2[e(G1 ) + e(G2 ) + e(G3 )] = 7
n
2
.
This completes the proof of Theorem 4.1 as desired. 5. Conclusions and open problem The key contribution of this paper is the following. We introduced the concept of k-decomposition for a graph, and then presented the Nordhaus–Gaddum-type inequality of a 3-decomposition of Kn for the hyper-Wiener index. However, exploring the corresponding Nordhaus–Gaddum-type inequality of a k-decomposition is still an open problem for k ≥ 4. We leave those questions for future research. Conjecture 5.1. Let (G1 , G2 , . . . , Gk ) be a k-decomposition of Kn such that each cell Gi is connected. Then for any sufficiently large n with respect to k, we have
n n+2 n (3k − 2) ≤ WW (G1 ) + WW (G2 ) + · · · + WW (Gk ) ≤ (k − 1) + + 2(k − 1)(n − 1). 2
The lower and upper bounds are sharp.
4
2
G. Su et al. / Theoretical Computer Science 471 (2013) 74–83
83
If Conjecture 5.1 holds, we can extend Theorem A for arbitrarily large k. The upper bound is attained when G1 = G2 =
· · · = Gk−1 = Pn , since if 2n − (k − 1)(n − 1) ≥ 21 nδ(Gk ) ≥ any sufficiently large n. Hence
n2 4
, i.e., k ≤ 4n , then by Lemma 2.1 we have diam(Gk ) = 2 for
WW (G1 ) + WW (G2 ) + · · · + WW (Gk ) = WW (G1 ) + WW (G2 ) + · · · + WW (Gk−1 ) + 3
n
2
− 2e(Gk )
n = [WW (G1 ) + 2e(G1 )] + [WW (G2 ) + 2e(G2 )] + · · · + [WW (Gk−1 ) + 2e(Gk−1 )] + 2 n = (k − 1)WW (Pn ) + 2(k − 1)(n − 1) + . 2
The lower bound is trivial and is attained n by any k-decomposition n(G1 , G2 , . . . , Gk ) of Kn with diam(Gi ) = 2 for k i = 1, 2, . . . , k, this is because WW (Gi ) = 3 2 − 2e(Gi ) and i=1 e(Gi ) 2 . Acknowledgments The authors thank the referees for their careful reading and valuable suggestions. This work was supported by Natural Science Fund of China (No. 11071016, 11171129) and Xinjiang Department of Education Scientific Research Projects in Universities (No. XJEDU2012I38). References [1] Z. An, B. Wu, D. Bin, Y. Wang, G. Su, Nordhaus–Gaddum-type theorem for diameter of graphs when decomposing into many parts, Discrete Math. Algorithms Appl. 3 (2011) 305–310. [2] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, American Elsevier [M], New York, 1976. [3] A.A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory and applications, Acta. Appl. Math. 66 (2001) 211–249. [4] P. Erdös, J. Pach, R. Pollack, Z. Tuza, Radius, diameter and minimum degree, J. Combin. Theory Ser. B 47 (1989) 73–79. [5] I. Gutman, W. Linert, I. Lukovits, A.A. Dobrynin, Trees with extremal hyper-Wiener index: mathematical basis and chemical, J. Chem. Inf. Comput. Sci. 37 (1997) 349–354. [6] M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, The hyper-Wiener index of graph operations, Comput. Math. Appl. 56 (2008) 1402–1407. [7] S. Klavžar, P. Zigert, I. Gutman, An algorithm for the calculation of the hyper-Wiener index of benzenoid hydrocarbons, Comput. Chem. 24 (2000) 229–233. [8] D.J. Klein, I. Lukovits, I. Gutman, On the definition of the hyper-Wiener index for cycle-containing structures, J. Chem. Inf. Comput. Sci. 35 (1995) 50–52. [9] X. Li, A.F. Jalbout, Bond order weighted hyper-Wiener index, J. Mol. Structure (Theochem) 634 (2003) 121–125. [10] D. Li, B. Wu, X. Yang, X. An, Nordhaus–Gaddum-type theorem for Wiener index of graphs when decomposing into three parts, Discrete Appl. Math. 159 (2011) 1594–1600. [11] W. Linert, I. Lukovits, Formulas for the hyper-Wiener and hyper-Detour indices of fused bicyclic structures, MATCH Comm. Math. Comput. Chem. 35 (1997) 65–74. [12] M. Liu, X. Tan, The first to (k + 1)-th smallest Wiener (hyper-Wiener) indices of connected graphs, Kragujevac J. Math. 32 (2009) 109–115. [13] D.E. Needham, I.C. Wei, P.G. Seybold, Molecular modeling of the physical properties of the alkanes, J. Amer. Chem. Soc. 110 (1988) 4186–4194. [14] E.A. Nordhaus, J.W. Gaddum, On complementary graphs, Amer. Math. Monthly 63 (1956) 175–177. [15] M. Randić, Novel molecular descriptor for structure-property studies, Chem. Phys. Lett. 211 (1993) 478–483. [16] G. Rücker, C. Rücker, On the topological indices, boiling points, and cycloalkanes, J. Chem. Inf. Comput. Sci. 39 (1999) 788–802. [17] G. Strang, Calculus, Wellesley-Cambridge Press, 1991. [18] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 (1947) 17–20. [19] W. Zhang, B. Wu, X. An, The hyper-Wiener index of the kth power of a graph, Discrete Math. Algor. Appl. 3 (2011) 17–23.
