Per-spectral and adjacency spectral characterizations of a complete ...

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Per-spectral and adjacency spectral characterizations of a complete graph removing six edges✩ Tingzeng Wu a,b , Heping Zhang a,∗ a

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, PR China

b

School of Mathematics and Statistics, Qinghai Nationalities University, Xining, Qinghai 810007, PR China

article

info

Article history: Received 27 November 2013 Received in revised form 30 August 2015 Accepted 14 September 2015 Available online xxxx Keywords: Characteristic polynomial Permanental polynomial Cospectral Permanental spectrum

abstract Cámara and Haemers (2014) investigated when a complete graph with some edges deleted is determined by its adjacency spectrum (DAS for short). They claimed: for any m ≥ 6 and every large enough n one can obtain graphs which are not DAS by removing m edges from a complete graph Kn . Let Gn denote the set of all graphs obtained from a complete graph Kn by deleting six edges. In this paper, we show that all graphs in Gn are uniquely determined by their permanental spectra. However, we show that for each n ≥ 7 or n = 5 there is just one pair of nonisomorphic cospectral graphs in Gn , and for n = 4 or 6 all graphs in Gn are DAS. © 2015 Elsevier B.V. All rights reserved.

1. Introduction All graphs considered in this work are undirected, finite and simple graphs. For notation and terminology not defined here, see [10]. Let G be a graph with the vertex set V (G) and the edge set E (G). Let A(G) be the adjacency matrix of G. The polynomial φ(G, x) = det(xI − A(G)), where I is the identity matrix, is called the characteristic polynomial of graph G. The adjacency spectrum of graph G consists of the eigenvalues of A(G) together with their multiplicities. Similarly, the permanental polynomial of G is defined as π (G, x) := per(xI − A(G)). The per-spectrum of graph G, denoted by ps(G), is the collection of all roots (together with their multiplicities) of π (G, x). Two graphs are cospectral (resp. per-cospectral) if they share the same adjacency spectrum (resp. per-spectrum). So two graphs are cospectral (resp. per-cospectral) if and only if they have the same characteristic (resp. permanental) polynomials. A graph G is determined by its adjacency spectrum (DAS for short) if every graph cospectral with G is isomorphic to G. Similarly, a graph G is said to be determined by its per-spectrum (DPS for short) if for any graph H, π (G, x) = π (H , x) implies that H is isomorphic to G. The characteristic polynomials of graphs and their applications are extensively examined (see for example [10]). However, only a few on the permanental polynomial and its potential applications have been published. Merris et al. [23] introduced the permanental polynomial of adjacency matrix of a general graph. The study of permanental polynomials of chemical molecular graphs were started by Kasum et al. [17]. Gutman and Cash [14] and Chen [7] obtained some relations between the coefficients of the permanental and characteristic polynomials of some chemical graphs, such as benzenoid

✩ Research supported by NSFC (11371180) and QHMU(2015XJZ12).

∗

Corresponding author. E-mail addresses: [email protected] (T. Wu), [email protected] (H. Zhang).

http://dx.doi.org/10.1016/j.dam.2015.09.014 0166-218X/© 2015 Elsevier B.V. All rights reserved.

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hydrocarbons, fullerenes, toroidal fullerenes and coronoid hydrocarbons. For more studies on permanental polynomials, see [1,5,6,9,18,27,34,35]. The spectrum of a graph encodes useful combinatorial information about the given graph, and the relationship between the structural properties of a graph. Which graphs are determined by the spectrum? In 1956 Günthard and Primas [13] first raised the question in a paper that relates the theory of graph spectra to Hückel’s theory from chemistry. van Dam and Haemers [30,31] gave an excellent survey of answers to the question of which graphs are determined by the spectra of some matrices associated to the graphs. In particular, the usual adjacency matrix was addressed. In the last decade, some types of graphs with very special structures have been proved to be DAS, such as the θ -graphs [25], dumbbell graphs without 4-cycle [32], lollipop graph [3,15], path and its complement [11], etc. See further Refs. [12,19]. Borowiecki [2] pointed out an interesting result: if G1 and G2 are bipartite graphs without cycles of length k, k ≡ 0 (mod 4), G1 and G2 are per-cospectral if and only if G1 and G2 are cospectral. Liu and Zhang in [20,21] investigated systemically which graphs are DPS. They found that graphs determined by the characteristic polynomial are not necessarily determined by the permanental polynomial, and showed that the complete graphs, stars, regular complete bipartite graphs, odd cycles and odd lollipop graphs are DPS. When restricting on connected graphs, the paths, even cycles C4l+2 (l ≥ 1), lollipop graphs Ln,2k+1 (k ≥ 1) and Ln,4 are DPS. In addition, Zhang et al. [36] showed that a graph obtained by removing five or fewer edges from a complete graph Kn is DPS, and proved that if X ⊆ E (Kn ) induces a star, a matching, or a disjoint union of a matching and a path P3 , then Kn − X is DPS. In [4], Cámara and Haemers showed that when at most five edges are deleted from Kn , there is just one pair of nonisomorphic cospectral graphs, and constructed nonisomorphic cospectral graphs for all n if six or more edges are deleted from Kn , provided that n is big enough. Motivated by these results, in this paper we intend to investigate when a complete graph Kn with six edges deleted is DPS and DAS respectively. An outline of this paper is as follows. In Section 2, we shall present some definitions and lemmas, and give a relation between the 4th coefficient of π (G, x) and the number of closed walks of length 4 of a graph G. In Section 3, we show that a complete graph with six edges deleted must be DPS. In Section 4, we prove that when six edges are deleted from Kn for n ≥ 7 and n = 5, there is just one pair of non-isomorphic cospectral graphs. We conclude with some discussions about potential applications and future research problems of permanental polynomials of chemical molecular graphs. 2. Preliminaries For graphs with six edges and isolated vertices, they have at most 12 vertices. We can see that there are exactly five non-isomorphic such graphs with at least 10 vertices. From Appendix I in [16] it can be seen that there exist exactly 63 nonisomorphic such graphs with at most nine vertices. So there exist 68 non-isomorphic graphs with six edges and no isolated vertices. For convenience, let Gn denote the set of all graphs obtained from Kn by deleting six edges. Thus, up to isomorphism there exist exactly 68 graphs in Gn for n ≥ 12, which are labeled by Gi (i = 1, 2, . . . , 68) and defined as illustrations in Fig. 1. Let ci (G) and pi (G) denote respectively the numbers  of i-cycles and i-vertex paths  in a graph G. Let K3 (G) be the set of all 3-cycles of G. For each K ∈ K3 (G), define dG (K ) = d (v) and D ( G ) = G v∈V (K ) K ∈K3 (G) dG (K ). For a subgraph H of G, let G − E (H ) be a subgraph obtained from G by deleting the edges of H. We present the following lemmas which are useful in the proof of the main results. Lemma 2.1 ([20,36]). n Let G ben−ai graph with n vertices and m edges, and let (d1 , d2 , . . . , dn ) be the degree sequence of G. Suppose that π (G, x) = . Then (i) b0 (G) = 1; (ii) b1 (G) = 0; (iii) b2 (G) = m; (iv) b3 (G) = −2c3 (G); (v) b4 (G) = i=0 bi (G)x

m 2

−

n di  i=1

2

+ 2c4 (G); (vi) b5 (G) = −2[c3 (G)(m + 3) − D(G) + c5 (G)].

Lemma 2.2 ([30,20]). The following parameters and properties of a graph G can be deduced from the per-spectrum and adjacency spectrum: (i) The number of vertices. (ii) The number of edges. (iii) The number of triangles. (iv) Whether G is bipartite. The following can be deduced from the per-spectrum of a graph G: (v) The length of a shortest odd cycle. (vi) The number of shortest odd cycles. The following can be deduced from the adjacency spectrum of a graph G: (vii) Whether G is regular. (viii) Whether G is regular with any fixed girth. (ix) The number of closed walks of any fixed length. Lemma 2.3 ([11]). Let H ⊆ Kn be a graph with l edges and let G = Kn − E (H ). Then c3 (G) =

n 3

− l(n − 2) +

  d(v)  v∈V (H )

2

− c3 (H ).

Using (1), we can compute the numbers of 3-cycles of all graphs in Gn ; see Table 1.

(1)

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Fig. 1. The graphs in Gn obtained from Kn by deleting six edges drawn lines in a disk.

By (i)–(iii) of Lemma 2.2 and Table 1, we immediately obtain the following result. Lemma 2.4. Graphs G1 , G2 , G17 and G58 are DPS and DAS. Lemma 2.5 ([36]). Let H ⊆ Kn be a graph with l edges and let G = Kn − E (H ). Then c4 (G) = 3

n 4

 − 2l

n−2



2

  d(v) 

+ (n − 5)

v∈V (H )

2

  +2

l

2

− p4 (H ) + c4 (H ).

(2)

By examining Table 1, we note that some graphs have the same number of 3-cycles. To obtain the main theorems in this paper, we calculate the number c4 of 4-cycles in these graphs by Eq. (2), as shown in Table 2. Lemma 2.6 ([36]). Let H ⊆ Kn have and let G = Kn − E (H ). Let dj (P3 ) denote the degree sum of three vertices on the  l edges  jth path P3 in H, and q = c5 (G) = 12

n 5



v∈V (H )

 − 6l

d(v) 2

n−2 3

. Then



 +4

n−4 1

   l

2

 − q + 2q



n−3

q   −2 (l + 2 − di (P3 ) + xH (P3 )) + p5 (H ) − c5 (H ), i =1

where xH (P3 ) = 1 if the ith P3 is contained in a triangle in H and 0 otherwise.

2



− p4 ( H )



n−4



1 (3)

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In order to compute b5 (G) by (vi) of Lemma 2.1, we need to know the values of c5 (G) and D(G) of some graphs in Gn . Using Eq. (3), we can obtain the number of 5-cycle of some graphs in Gn as in Table 3. Furthermore, we also calculate D(G) for the above graphs as follows. Lemma 2.7.

 n   (3n − 3) − 24n2 + 99n − 54,   3       n (3n − 3) − 24n2 + 99n − 51     3    n   (3n − 3) − 24n2 + 104n − 72    3    n    (3n − 3) − 24n2 + 104n − 71   3  n     (3n − 3) − 24n2 + 114n − 105   3     n  2    3 (3n − 3) − 24n + 116n − 114      n    (3n − 3) − 24n2 + 104n − 70   3   n     (3n − 3) − 24n2 + 109n − 86    3   n     (3n − 3) − 24n2 + 114n − 104   3     n   (3n − 3) − 24n2 + 109n − 88   3   n  (3n − 3) − 24n2 + 114n − 96 D(G) =  3  n     (3n − 3) − 24n2 + 119n − 120   3      n (3n − 3) − 24n2 + 119n − 122     3    n   (3n − 3) − 24n2 + 116n − 112    3  n     (3n − 3) − 24n2 + 114n − 105   3       n (3n − 3) − 24n2 + 114n − 101      3  n  2    3 (3n − 3) − 24n + 114n − 103   n     (3n − 3) − 24n2 + 109n − 85   3    n    (3n − 3) − 24n2 + 104n − 68    3    n    (3n − 3) − 24n2 + 114n − 102   3      n (3n − 3) − 24n2 + 126n − 150 3

if G = G4 with n ≥ 10, if G = G7 with n ≥ 9, if G = G11 with n ≥ 8, if G = G15 with n ≥ 9, if G = G19 with n ≥ 7, if G = G22 with n ≥ 7, if G = G23 with n ≥ 8, if G = G24 with n ≥ 8, if G = G25 with n ≥ 8, if G = G26 with n ≥ 8, if G = G38 with n ≥ 8, if G = G40 with n ≥ 6, if G = G44 with n ≥ 6, if G = G52 with n ≥ 6, if G = G60 with n ≥ 7, if G = G62 with n ≥ 7, if G = G65 with n ≥ 7, if G = G28 or G63 with n ≥ 7, if G = G31 or G36 with n ≥ 8, if G = G13 (G39 ) with n ≥ 9 (n ≥ 6), if G = G41 (G50 ) with n ≥ 5 (n ≥ 6).

The proof is not difficult but too long to give here. We only give the proof of the case G = G44 . Proof. We use the notation of graph G44 in Fig. 1 and let H denote a subgraph of Kn obtained by joining a vertex of cycle C4 to an end vertex of a path P2 . Let e1 = v1 v  2 , e2 = v2 v3 , e3 = v3 v4 , e4 = v4 v1 , e5 = v4 v5 and e6 = v5 v6 denote the edges of n H. Direct computation yields D(Kn ) = 3 (3n − 3). We will compute D(G44 ) by removing one edge of the edges e1 , . . . , e6 at a time. Step 1: We observe  that Kn has n − 2 triangles containing e1 , and these triangles will be destroyed in Kn − e1 . We also note that Kn − e1 has 2

n −2 2

triangles containing exactly one endpoint of e1 . For each of such 3-cycles, its degree sum in Kn − e1

will decrease by 1. As each triangle in Kn has degree sum 3n − 3, it follows that D(Kn ) − D(Kn − e1 ) = (n − 2)(3n − 3) + 2



n−2 2



= 4n2 − 14n + 12.

(4)

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Table 1 The number c3 (G) of triangles of all graphs in Gn . Graphs

c3 (G)

G1

n  3n 

G2

3 n 3 n 3 n 3 n 3 n 3 n 3 n 3 n 3 n 3 n 3 n 3

− 6n + 12 − 6n + 13

G17

 

− 6n + 22

G58

 

− 6n + 27

G55 , G66

 

− 6n + 23

G3 , G5 , G6

 

− 6n + 14

G4 , G7 , G8 , G10 , G14

 

− 6n + 15

G43 , G46 , G54 , G57 , G67

 

− 6n + 21

G9 , G11 , G12 , G15 , G23 , G31 , G34 , G36 , G37

 

− 6n + 16

G20 , G27 , G30 , G40 , G44 , G51 , G53 , G61 , G64

 

− 6n + 19

G16 , G21 , G24 , G26 , G28 , G29 , G32 , G33 , G35 , G63

 

− 6n + 17

G41 , G42 , G45 , G47 , G48 , G49 , G50 , G56 , G59 , G68

 

− 6n + 20

G13 , G18 , G19 , G22 , G25 , G38 , G39 , G52 , G60 , G62 , G65

 

− 6n + 18

Table 2 The number of quadrangles of some graphs in Gn . Graph G3 G8 , G9 G11 , G23 G12 , G32 G19 , G65 G20 , G59 G22 , G52 G24 , G28 G26 , G63 G34 , G35 G39 , G60 G40 , G44 G43 , G56 G45 , G51 G53 , G68 G4 , G5 , G7 G15 , G31 , G36 G21 , G25 , G62 G41 , G50 , G57 G13 , G33 , G37 , G38

c4 (G) 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

n  4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n  4

2

− 6n + 32n − 16 2

− 6n + 34n − 26 2

− 6n + 34n − 29 2

− 6n + 35n − 33 2

− 6n + 36n − 41 2

− 6n + 38n − 50 2

− 6n + 37n − 46 2

− 6n + 35n − 34 2

− 6n + 35n − 35 2

− 6n + 35n − 32 2

− 6n + 36n − 42 2

− 6n + 37n − 48 2

− 6n + 39n − 58 2

− 6n + 38n − 53 2

− 6n + 38n − 52 2

− 6n + 33n − 21 2

− 6n + 34n − 28 2

− 6n + 36n − 40 2

− 6n + 39n − 60

Graph

c4 (G)

G6

3

G10

3

G14

3

G16

3

G18

3

G27

3

G29

3

G30

3

G42

3

G46

3

G47

3

G48

3

G49

3

G54

3

G55

3

G61

3

G64

3

G66

3

G67

3

n  4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n   4n  4

− 6n2 + 32n − 17 − 6n2 + 33n − 22 − 6n2 + 33n − 23 − 6n2 + 35n − 36 − 6n2 + 38n − 51 − 6n2 + 37n − 44 − 6n2 + 36n − 38 − 6n2 + 37n − 41 − 6n2 + 38n − 54 − 6n2 + 41n − 70 − 6n2 + 40n − 65 − 6n2 + 42n − 71 − 6n2 + 40n − 62 − 6n2 + 40n − 64 − 6n2 + 42n − 72 − 6n2 + 37n − 47 − 6n2 + 37n − 45 − 6n2 + 41n − 65 − 6n2 + 39n − 57

2

− 6n + 36n − 36

Step 2: Note that Kn − e1 has n − 3 triangles containing e2 , and these triangles will bedestroyed in Kn − {e1 , e2 }, and that the  degree sum of each such triangle in Kn − e1 is 3n − 4. We also note that Kn − e1 has 2

n −3 2

triangles containing exactly one

endpoint of e2 , and for each of such 3-cycles, its degree sum in Kn − {e1 , e2 } will decrease by 1 from its degree sum in Kn − e1 . Moreover, Kn − e1 has n − 3 triangles containing edge v1 v3 , and for each of such 3-cycles, its degree sum in Kn − {e1 , e2 } will decrease by 1 from its degree sum in Kn − e1 . Thus, after deleting e2 in Kn − e1 , we have D(Kn − e1 ) − D(Kn − e1 − e2 ) = (n − 3)(3n − 4) + 2



n−3 2



+ (n − 3) = 4n2 − 19n + 21.

(5)

Step 3: Again Kn − e1 − e2 has n − 3 triangles containing e3 . Among these triangles, v1 v3 v4 has degree sum 3n − 5 in Kn − e1 − e2 , and each of the other 3-cycles has degree   sum 3n − 4 in Kn − e1 − e2 . All these 3-cycles will be destroyed in Kn − {e1 , e2 , e3 , e4 }. Moreover, Kn − e1 − e2 has 2

n−4 2

3-cycles each of which contains exactly one endpoint of e3 and two

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Table 3 The number of pentagons of some graphs in Gn . Graph G4 G11 G15 G22 G24 G26 G31 G38 G40 G44 G52 G62 G65

c5 (G) 12 12 12 12 12 12 12 12 12 12 12 12 12

n  5n   5n   5n   5n   5n   5n   5n   5n   5n   5n   5n   5n  5

3

2

3

2

3

2

3

2

3

2

3

2

3

2

3

2

3

2

3

2

3

2

3

2

3

2

− 6n + 57n − 129n − 30 − 6n + 58n − 144n + 16 − 6n + 58n − 142n + 6 − 6n + 61n − 178n + 103 − 6n + 59n − 154n + 36 − 6n + 59n − 155n + 43 − 6n + 58n − 142n + 1 − 6n + 60n − 162n + 36 − 6n + 61n − 180n + 112 − 6n + 61n − 181n + 118 − 6n + 61n − 178n + 100 − 6n + 60n − 166n + 65

c5 (G)

G7

12

G13

12

G19

12

G23

12

G25

12

G28

12

G36

12

G39

12

G41

12

G50

12

G60

12

G63

12

n  5n   5n   5n   5n   5n   5n   5n   5n   5n   5n   5n  5

− 6n3 + 57n2 − 129n − 36 − 6n3 + 60n2 − 162n + 48 − 6n3 + 60n2 − 168n + 80 − 6n3 + 58n2 − 143n + 10 − 6n3 + 60n2 − 166n + 68 − 6n3 + 59n2 − 155n + 36 − 6n3 + 58n2 − 142n − 6n3 + 60n2 − 168n + 78 − 6n3 + 63n2 − 205n + 188 − 6n3 + 63n2 − 204n + 183 − 6n3 + 60n2 − 168n + 81 − 6n3 + 59n2 − 155n + 39

− 6n + 60n − 167n + 73

vertices in V (Kn ) − {v1 , v2 , v3 , v4 }; and 3 The degree sum of each of these 2

Graph



n −4 2





n−4 1



+3



3-cycles each of which contains exactly one of edges in {v1 v3 , v1 v4 , v2 v4 }.

n −4 1



triangles in Kn − {e1 , e2 } will be decreased by 1 in Kn − {e1 , e2 , e3 }. Thus

D(Kn − e1 − e2 ) − D(Kn − e1 − e2 − e3 )

= (3n − 5) + (n − 4)(3n − 4) + 2



n−4



 +3

2

n−4



1

= 4n2 − 19n + 19.

(6)

Step 4: We again note that Kn − e1 − e2 − e3 has n − 4 triangles containing e4 , and the degree sum of each of these triangle in Kn − − e2 − e3 is 3n − 5. All these 3-cycles will be destroyed in Kn − {e1 , e2 , e3 , e4 }. Furthermore, Kn − e1 − e2 − e3  e1  n −4

has 2

2

3-cycles each of which contains exactly one endpoint of e4 and two vertices in V (Kn ) − {v1 , v2 , v3 , v4 }; and has

  + 2 n−1 4 triangles in Kn − {e1 , e2 , e3 } will be decreased by 1 in Kn − {e1 , e2 , e3 , e4 }. Thus, after deleting e4 in Kn − e1 − e2 − e3 , we 2



n−4 1

3-cycles each of which contains exactly edge in {v1 v3 , v2 v4 }. The degree sum of each of these 2



n−4 2



have D(Kn − e1 − e2 − e3 ) − D(Kn − e1 − e2 − e3 − e4 )

= (n − 4)(3n − 5) + 2



n−4



2

 +2

n−4



1

= 4n2 − 24n + 32.

(7)

Step 5: We observe that Kn − e1 − e2 − e3 − e4 has n − 4 triangles containing e5 . Among these triangles, v2 v4 v5 has degree sum 3n − 7, and n − 5 others each has degree sum 3n − 5 in Kn − e1 − e2 − e3 − e4 . All these 3-cycles will be destroyed in Kn − e1 − e2 − e3 − e4 − e5 .   Moreover, Kn − e1 − e2 − e3 − e4 has 2

of V (Kn ) − {v1 , v2 , v3 , v4 , v5 }, has 4



n −5 1

n −5

2

3-cycles each of which contains exactly one endpoint of e5 and two vertices

triangles each of which contains one of edges in {v1 v5 , v2 v5 , v3 v5 , v2 v4 } and

a vertex (Kn ) −  in V   {v1 , v2 , v3 , v4 , v5 }, plus the 3-cycles v1 v3 v5 . By direct computation, the sum degree of each of these n−5 n −5 2 2 + 4 1 + 1 in Kn − {e1 , e2 , e3 , e4 } will be decreased by 1 in Kn − {e1 , e2 , e3 , e4 , e5 }. Thus, after deleting e5 in Kn − {e1 , e2 , e3 , e4 }, we have

D(Kn − e1 − e2 − e3 − e4 ) − D(Kn − e1 − e2 − e3 − e4 − e5 )

= (3n − 7) + (n − 5)(3n − 5) + 2



n−5 2



 +4

n−5 1



+ 1 = 4n2 − 24n + 29.

(8)

Step 6: Similarly, we note that Kn − e1 − e2 − e3 − e4 − e5 has n − 3 triangles containing e6 . Among these triangles, each of v1 v5 v6 , v2 v5 v6 and v3 v5 v6 has degree sum 3n−6, and each of the n−6 others has degree sum 3n−4 in Kn −e1 −e2 −e3 −e4 −e5 . All these 3-cycles will be destroyed in Kn − e1 − e2 − e 3 − e 4 − e5 − e6 . Moreover, Kn − e1 − e2 − e3 − e4 − e5 has 2

n−6 2

3-cycles each of which contains exactly one endpoint of

e6 and two vertices of V (Kn ) − {v1 , v2 , v3 , v4 , v5 , v6 }, and 7



n −6 1



3-cycles each of which contains one of edges in

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{v1 v5 , v2 v5 , v3 v5 , v1 v6 , v2 v6 , v3 v6 , v4 v6 } and a vertex in V (Kn )−{v1 , v2 , v3 , v4, v5 ,  v6 }, plus  thethree other 3-cycles v1 v3 v5 , n −6 v1 v3 v6 , and v2 v4 v6 . By direct computation, the sum degree of each of these 2 2 + 7 n−1 6 + 3 will be decreased by 1 in Kn − {e1 , e2 , e3 , e4 , e5 , e6 }. It follows that D(Kn − e1 − e2 − e3 − e4 − e5 ) − D(Kn − e1 − e2 − e3 − e4 − e5 − e6 )

= 3(3n − 6) + (n − 6)(3n − 4) + 2 Combining Eqs. (4)–(9), we obtain D(G44 ) =



n−6



2

n 3

 +7

n−6



1

+ 3 = 4n2 − 19n + 9.

(3n − 3) − 24n2 + 119n − 122.

(9)



Let NG (H ) be the number of subgraphs of a graph G which are isomorphic to H and let NG (i) denote the number of closed walks of length i of G. Lemma 2.8 ([24]). The numbers of closed walks of lengths 4 and 5 of a graph G with m edges are as follows. (i) NG (4) = 2m + 4NG (P3 ) + 8NG (C4 ), (ii) NG (5) = 30NG (K3 ) + 10NG (C5 ) + 10NG (K1,3 + e). For a graph G, it follows that NG (P3 ) = between NG (4) and b4 (G).

n

i=1,vi ∈V (G)



d(vi ) 2



. By Lemmas 2.1 and 2.8, we can deduce the following relation

Corollary 2.9. NG (4) = 2m2 + 16c4 (G) − 4b4 (G). Lemma 2.10. NG (K1,3 + e) = D(G) − 6c3 (G). Proof. For each K ∈ K3 (G), it follows that the number of K1,3 + e in G equals dG (K ) − 6. By Eq. (1) and direct computation, we have NG (K1,3 + e) =

c3 (G)

i=1,Ki ∈K3 (G)

(dG (Ki ) − 6) = D(G) − 6c3 (G).



3. Each graph in Gn is determined by its per-spectrum In this section, our aim is to obtain the following main theorem. Theorem 3.1. All graphs in Gn are DPS. Before proving Theorem 3.1, we first outline an idea. By Table 1, Gn is partitioned into different groups according to the number of triangles which a graph in Gn will have. From Lemma 2.2 we may see that two graphs in different groups are not per-cospectral since they have different number of triangles. Further, we assume all graphs in Gn have the same number of vertices and edges. By Lemma 2.1 and Tables 1–3, we calculate the fourth and fifth coefficients of the permanental polynomials of graphs in each group above, respectively, and compare the coefficients to determine whether such graphs are per-cospectral or not. To complete the proof it is sufficient to verify each statement of the following lemma. Lemma 3.2. Each of the following holds. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

Graphs G3 , G5 and G6 are not pairwise per-cospectral. Graphs G55 and G66 are not per-cospectral. Graphs G4 , G7 , G8 , G10 and G14 are not pairwise per-cospectral. Graphs G43 , G46 , G54 , G57 and G67 are not pairwise per-cospectral. Graphs G9 , G11 , G12 , G15 , G23 , G31 , G34 , G36 and G37 are not pairwise per-cospectral. Graphs G20 , G27 , G30 , G40 , G44 , G51 , G53 , G61 and G64 are not pairwise per-cospectral. Graphs G16 , G21 , G24 , G26 , G28 , G29 , G32 , G33 , G35 and G63 are not pairwise per-cospectral. Graphs G41 , G42 , G45 , G47 , G48 , G49 , G50 , G56 , G59 and G68 are not pairwise per-cospectral. Graphs G13 , G18 , G19 , G22 , G25 , G38 , G39 , G52 , G60 , G62 and G65 are not pairwise per-cospectral.

Proof. (i) By (v) of Lemma 2.1 and Table 2, we have b4 (G3 ) − b4 (G5 ) =

i =1 (



di (G5 ) 2





di (G3 ) 2



) + 2(c4 (G3 ) − c4 (G5 )) = −2n + 11 ̸= 0. Hence G3 and G5 are not per-cospectral. Similarly, by the same argument as above, we have b4 (G3 )− b4 (G6 ) = 2 and b4 (G5 ) − b4 (G6 ) = 2n − 9, which imply that G6 is not per-cospectral with either G3 or G5 . (ii) By (v) of Lemma 2.1 and Table 2, we have b4 (G55 ) − b4 (G66 ) = 2n − 15 ̸= 0. This implies that G55 and G66 are not n

−

per-cospectral. (iii) By Lemmas 2.1 and 2.7 and Table 3, we obtain that b5 (G4 ) − b5 (G7 ) = −2[(D(G7 ) − D(G4 )) + (c5 (G4 ) − c5 (G7 ))] = −18 ̸= 0, which implies that G4 is not per-cospectral with G7 . In addition, by (v) of Lemma 2.1 and Table 2, we have b4 (G4 ) − b4 (G8 ) = −2n + 11, b4 (G4 ) − b4 (G10 ) = 2, b4 (G4 ) − b4 (G14 ) = 4, b4 (G7 ) − b4 (G8 ) = −2n + 11, b4 (G7 ) − b4 (G10 ) = 2, b4 (G7 ) − b4 (G14 ) = 4, b4 (G8 ) − b4 (G10 ) = 2n − 9, b4 (G8 ) − b4 (G14 ) = 2n − 7 and b4 (G10 ) − b4 (G14 ) = 2. These imply that no pairs of G4 , G7 , G8 , G10 and G14 are per-cospectral.

8

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(iv) By (v) of Lemma 2.1 and Table 2, we have b4 (G46 ) − b4 (G67 ) = 4n − 28. It is easy to see that b4 (G46 ) − b4 (G67 ) = 0 only when n = 7. Employing Maple 12.0 with n = 7, we compute the permanental polynomials of G46 and G67 , respectively. We found that π (G46 , x) = x7 + 15x5 − 28x4 + 105x3 − 166x2 + 171x − 42 and π (G67 , x) = x7 + 15x5 − 28x4 + 105x3 − 174x2 + 211x − 114. So π (G46 , x) ̸= π (G67 , x). These indicate that G46 and G67 are not per-cospectral. Analogously, by (v) of Lemma 2.1 and Table 2, we have b4 (G43 ) − b4 (G46 ) = −4n + 26, b4 (G43 ) − b4 (G54 ) = −2n + 13, b4 (G43 ) − b4 (G57 ) = 4, b4 (G43 ) − b4 (G67 ) = −2, b4 (G46 ) − b4 (G54 ) = 2n − 13, b4 (G46 ) − b4 (G57 ) = 4n − 22, b4 (G54 ) − b4 (G57 ) = 2n − 9, b4 (G54 ) − b4 (G67 ) = 2n − 15 and b4 (G57 ) − b4 (G67 ) = −6. We conclude that any two graphs in {G43 , G46 , G54 , G57 , G67 } are not per-cospectral. (v) Assume that G11 and G23 are cospectral. By Lemmas 2.1 and 2.7 and Table 3, we have 0 = b5 (G11 )− b5 (G23 ) = 2n − 16, forcing n = 8. Using Maple 12.0 with n = 8, we calculate the permanental polynomials of G11 and G23 , respectively. We discovered that π (G11 , x) = x8 + 22x6 − 48x5 + 269x4 − 752x3 + 1884x2 − 2880x + 2276 and π (G23 , x) = x8 + 22x6 − 48x5 + 269x4 − 752x3 + 1857x2 − 2908x + 2324. So, π (G11 , x) ̸= π (G23 , x), a contradiction, which means that G11 is not per-cospectral with G23 . Similarly, we have b5 (G15 ) − b5 (G31 ) = −16, b5 (G15 ) − b5 (G36 ) = −18 and b5 (G31 ) − b5 (G36 ) = −2. These imply that none of pairs G15 and G31 , G15 and G36 , and G31 and G36 are per-cospectral. By (v) of Lemma 2.1 and Table 2, we obtain that b4 (G11 ) − b4 (G37 ) = −4n + 16 and b4 (G23 ) − b4 (G37 ) = −4n + 16. Hence b4 (G11 ) − b4 (G37 ) = 0 and b4 (G23 ) − b4 (G37 ) = 0 only when n = 4. However, by the definitions of G11 , G23 and G37 , we note that |V (G11 )| ≥ 8, |V (G23 )| ≥ 8 and |V (G37 )| ≥ 6. These imply that neither pair G11 and G37 nor G23 and G37 are per-cospectral. Similarly, by (v) of Lemma 2.1 and Table 2, we also have b4 (G9 ) − b4 (G11 ) = 6, b4 (G9 ) − b4 (G12 ) = −2n + 15, b4 (G9 ) − b4 (G15 ) = 4, b4 (G9 ) − b4 (G23 ) = 6, b4 (G9 ) − b4 (G31 ) = 4, b4 (G9 ) − b4 (G34 ) = −2n + 13, b4 (G9 ) − b4 (G36 ) = 4, b4 (G9 ) − b4 (G37 ) = −4n + 22, b4 (G11 ) − b4 (G12 ) = −2n + 9, b4 (G11 ) − b4 (G15 ) = −2, b4 (G11 ) − b4 (G31 ) = −2, b4 (G11 ) − b4 (G34 ) = −2n + 7, b4 (G11 ) − b4 (G36 ) = −2, b4 (G12 ) − b4 (G15 ) = 2n − 11, b4 (G12 ) − b4 (G23 ) = 2n − 9, b4 (G12 ) − b4 (G31 ) = 2n − 11, b4 (G12 ) − b4 (G34 ) = −2, b4 (G12 ) − b4 (G36 ) = 2n − 11, b4 (G12 ) − b4 (G37 ) = −2n + 7, b4 (G15 ) − b4 (G23 ) = 2, b4 (G15 ) − b4 (G34 ) = −2n + 9, b4 (G15 ) − b4 (G37 ) = −4n + 18, b4 (G23 ) − b4 (G31 ) = −2, b4 (G23 ) − b4 (G34 ) = −2n + 7, b4 (G23 ) − b4 (G36 ) = −2, b4 (G31 ) − b4 (G34 ) = −2n + 9, b4 (G31 ) − b4 (G37 ) = −4n + 18, b4 (G34 ) − b4 (G36 ) = 2n − 9, b4 (G34 ) − b4 (G37 ) = −2n + 9 and b4 (G36 ) − b4 (G37 ) = −4n + 18. So we conclude that no pairs of G9 , G11 , G12 , G15 , G23 , G31 , G34 , G36 and G37 are per-cospectral. (vi) We assume that G40 and G44 are per-cospectral. By Lemmas 2.1 and 2.7 and Table 3, we have 0 = b5 (G40 )− b5 (G44 ) = −2n + 16, forcing n = 8. Employing Maple 12.0 with n = 8, we compute π (G40 , x) and π (G44 , x), respectively. We found that π (G40 , x) = x8 + 22x6 − 54x5 + 276x4 − 758x3 + 1719x2 − 2516x + 1892 and π (G44 , x) = x8 + 22x6 − 54x5 + 276x4 − 758x3 + 1740x2 − 2536x + 1821. So π (G40 , x) ̸= π (G44 , x), a contradiction, which indicates that G40 and G44 cannot be per-cospectral. Additionally, by (v) of Lemma 2.1 and Table 2, we have b4 (G20 ) − b4 (G27 ) = 2n − 13, b4 (G20 ) − b4 (G30 ) = 2n − 19, b4 (G20 ) − b4 (G40 ) = 2n − 5, b4 (G20 ) − b4 (G44 ) = 2n − 5, b4 (G20 ) − b4 (G51 ) = 6, b4 (G20 ) − b4 (G53 ) = 4, b4 (G20 ) − b4 (G61 ) = 2n − 7, b4 (G20 ) − b4 (G64 ) = 2n − 11, b4 (G27 ) − b4 (G30 ) = −6, b4 (G27 ) − b4 (G40 ) = 8, b4 (G27 ) − b4 (G44 ) = 8, b4 (G27 ) − b4 (G51 ) = −2n + 19, b4 (G27 ) − b4 (G53 ) = −2n + 17, b4 (G27 ) − b4 (G61 ) = 6, b4 (G27 ) − b4 (G64 ) = 2, b4 (G30 ) − b4 (G40 ) = 14, b4 (G30 ) − b4 (G44 ) = 14, b4 (G30 ) − b4 (G51 ) = −2n + 25, b4 (G30 ) − b4 (G53 ) = −2n + 23, b4 (G30 ) − b4 (G61 ) = 12, b4 (G30 ) − b4 (G64 ) = 8, b4 (G40 ) − b4 (G51 ) = −2n + 11, b4 (G40 ) − b4 (G53 ) = −2n + 9, b4 (G40 )− b4 (G61 ) = −2, b4 (G40 )− b4 (G64 ) = −6, b4 (G51 )− b4 (G53 ) = −2, b4 (G51 )− b4 (G61 ) = 2n − 13, b4 (G51 )− b4 (G64 ) = 2n − 17, b4 (G53 ) − b4 (G61 ) = 2n − 11, b4 (G53 ) − b4 (G64 ) = 2n − 15 and b4 (G61 ) − b4 (G64 ) = −4. These imply that no pairs of G20 , G27 , G30 , G40 , G44 , G51 , G53 , G61 and G64 are per-cospectral. (vii) By Lemmas 2.1 and 2.7 and Table 3, we have b5 (G24 ) − b5 (G28 ) = −2n − 2. Hence b5 (G24 ) − b5 (G28 ) = 0 only when n = −1. This contradicts n ≥ 0. So G24 and G28 are not per-cospectral. Similarly, by the same argument as above, we also have b5 (G26 ) − b5 (G63 ) = −14 ̸= 0, which implies that G26 is not per-cospectral with G63 . In addition, by (v) of Lemma 2.1 and Table 2, we obtain that b4 (G16 ) − b4 (G21 ) = −2n + 9, b4 (G16 ) − b4 (G24 ) = −4, b4 (G16 )− b4 (G26 ) = −2, b4 (G16 )− b4 (G28 ) = −4, b4 (G16 )− b4 (G29 ) = −2n + 5, b4 (G16 )− b4 (G32 ) = −6, b4 (G16 )− b4 (G33 ) = −2n + 1, b4 (G16 ) − b4 (G35 ) = −8, b4 (G16 ) − b4 (G63 ) = −2, b4 (G21 ) − b4 (G24 ) = 2n − 13, b4 (G21 ) − b4 (G26 ) = 2n − 11, b4 (G21 ) − b4 (G28 ) = 2n − 13, b4 (G21 ) − b4 (G29 ) = −4, b4 (G21 ) − b4 (G32 ) = 2n − 15, b4 (G21 ) − b4 (G33 ) = −8, b4 (G21 ) − b4 (G35 ) = 2n − 17, b4 (G21 ) − b4 (G63 ) = 2n − 11, b4 (G24 ) − b4 (G26 ) = 2, b4 (G24 ) − b4 (G29 ) = −2n + 9, b4 (G24 ) − b4 (G32 ) = −2, b4 (G24 ) − b4 (G33 ) = −2n + 5, b4 (G24 ) − b4 (G35 ) = −4, b4 (G24 ) − b4 (G63 ) = 2, b4 (G26 ) − b4 (G28 ) = −2, b4 (G26 ) − b4 (G29 ) = −2n + 7, b4 (G26 ) − b4 (G32 ) = −4, b4 (G26 ) − b4 (G33 ) = −2n + 3, b4 (G26 ) − b4 (G35 ) = −6, b4 (G28 ) − b4 (G29 ) = −2n + 9, b4 (G28 ) − b4 (G32 ) = −2, b4 (G28 ) − b4 (G33 ) = −2n + 5, b4 (G28 ) − b4 (G35 ) = −4, b4 (G28 )−b4 (G63 ) = 2, b4 (G29 )−b4 (G32 ) = 2n−11, b4 (G29 )−b4 (G33 ) = −4, b4 (G29 )−b4 (G35 ) = 2n−13, b4 (G29 )−b4 (G63 ) = 2n − 7, b4 (G32 ) − b4 (G33 ) = −2n + 7, b4 (G32 ) − b4 (G35 ) = −2, b4 (G32 ) − b4 (G63 ) = 4, b4 (G33 ) − b4 (G35 ) = 2n − 9, b4 (G33 ) − b4 (G63 ) = 2n − 3 and b4 (G35 ) − b4 (G63 ) = 6. So we conclude that any two graphs in {G16 , G21 , G24 , G26 , G28 , G29 , G32 , G33 , G35 , G63 } are not per-cospectral. (viii) By Lemmas 2.1 and 2.7 and Table 3, we have b5 (G41 ) − b5 (G50 ) = 2n − 10. However, by the definition of G50 , it can be seen that |V (G50 )| ≥ 6. This means that G41 and G50 are not per-cospectral. By (v) of Lemma 2.1 and Table 2, we have b4 (G42 ) − b4 (G47 ) = −4n + 24. Hence b4 (G42 ) − b4 (G47 ) = 0 only when n = 6. Using Maple 12.0 with n = 6, we compute the permanental polynomials of G42 and G47 , respectively. We found that

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π(G42 , x) = x6 + 9x4 − 8x3 + 22x2 − 20x + 13 and π (G47 , x) = x6 + 9x4 − 8x3 + 22x2 − 12x + 4. So π (G42 , x) ̸= π (G47 , x), which implies that G42 and G47 are not per-cospectral. Similarly, by (v) of Lemma 2.1 and Table 2, we note that b4 (G47 )− b4 (G59 ) = 4n − 32. It is obvious that b4 (G47 )− b4 (G59 ) = 0 if and only if n = 8. Employing Maple 12.0 with n = 8, we have π (G47 , x) = x8 + 22x6 − 56x5 + 287x4 − 764x3 + 1638x2 − 2056x + 1140 and π (G59 , x) = x8 + 22x6 − 56x5 + 287x4 − 782x3 + 1734x2 − 2398x + 1656. These results indicate that G47 is not per-cospectral with G59 . By the same argument as above, we again note that b4 (G47 ) − b4 (G68 ) = 4n − 28. Using Maple 12.0 with n = 7, we have π (G47 , x) = x7 + 15x5 − 26x4 + 102x3 − 160x2 + 178x − 72 and π (G68 , x) = x7 + 15x5 − 26x4 + 99x3 − 168x2 + 202x − 90. So G47 and G68 are not per-cospectral. Further, by (v) of Lemma 2.1 and Table 2, we obtain that b4 (G48 ) − b4 (G49 ) = 4n − 20. Hence b4 (G48 ) − b4 (G49 ) = 0 only when n = 5. Employing Maple 12.0 with n = 5, we have π (G48 , x) = x5 + 4x3 and π (G49 , x) = x5 + 4x3 + 4x. So π(G48 , x) ̸= π (G49 , x), which indicates that G48 and G49 are not per-cospectral. Analogously, by the same argument as above, we have b4 (G45 ) − b4 (G49 ) = −4n + 20. It is easy to see that there only exists n = 5 such that b4 (G45 ) − b4 (G49 ) = 0. This contradicts |V (G45 )| ≥ 6. Then G45 and G49 are not per-cospectral. Finally, by (v) of Lemma 2.1 and Table 2, we have b4 (G41 ) − b4 (G42 ) = 2n − 13, b4 (G41 ) − b4 (G45 ) = 2n − 15, b4 (G41 ) − b4 (G47 ) = −2n + 11, b4 (G41 ) − b4 (G48 ) = −6n + 25, b4 (G41 ) − b4 (G49 ) = −2n + 5, b4 (G41 ) − b4 (G56 ) = −4, b4 (G41 ) − b4 (G59 ) = 2n − 21, b4 (G41 ) − b4 (G68 ) = 2n − 17, b4 (G42 ) − b4 (G45 ) = −2, b4 (G42 ) − b4 (G48 ) = −8n + 38, b4 (G42 ) − b4 (G49 ) = −4n + 18, b4 (G42 ) − b4 (G50 ) = −2n + 13, b4 (G42 ) − b4 (G56 ) = −2n + 9, b4 (G42 ) − b4 (G59 ) = −8, b4 (G42 ) − b4 (G68 ) = −4, b4 (G45 ) − b4 (G47 ) = −4n + 26, b4 (G45 ) − b4 (G48 ) = −6, b4 (G45 ) − b4 (G50 ) = −2n + 15, b4 (G45 ) − b4 (G56 ) = −2n + 11, b4 (G45 ) − b4 (G59 ) = −6, b4 (G45 ) − b4 (G68 ) = −2, b4 (G47 ) − b4 (G48 ) = −4n + 14, b4 (G47 ) − b4 (G49 ) = −6, b4 (G47 ) − b4 (G50 ) = 2n − 11, b4 (G47 ) − b4 (G56 ) = 2n − 15, b4 (G48 ) − b4 (G50 ) = 6n − 25, b4 (G48 ) − b4 (G56 ) = 6n − 29, b4 (G48 ) − b4 (G59 ) = 8n − 46, b4 (G48 ) − b4 (G68 ) = 8n − 42, b4 (G49 ) − b4 (G50 ) = 2n − 5, b4 (G49 ) − b4 (G56 ) = 2n − 9, b4 (G49 ) − b4 (G59 ) = 4n − 26, b4 (G49 ) − b4 (G68 ) = 4n − 22, b4 (G50 ) − b4 (G56 ) = −4, b4 (G50 ) − b4 (G59 ) = 2n − 21, b4 (G50 ) − b4 (G68 ) = 2n − 17, b4 (G56 ) − b4 (G59 ) = 2n − 17, b4 (G56 ) − b4 (G68 ) = 2n − 13, and b4 (G59 ) − b4 (G68 ) = 4. It follows by similar argument that no pairs of G41 , G42 , G45 , G47 , G48 , G49 , G50 , G56 , G59 and G68 are per-cospectral. (ix) By Lemmas 2.1 and 2.7 and Table 3, we have b5 (G19 ) − b5 (G65 ) = 2n − 18. It is clear that b5 (G19 ) − b5 (G65 ) = 0 only when n = 9. Employing Maple 12.0 with n = 9, we have π (G40 , x) = x9 + 30x7 − 96x6 + 611x5 − 2314x4 + 7670x3 − 18418x2 + 29388x − 23000 and π (G44 , x) = x9 + 30x7 − 96x6 + 611x5 − 2314x4 + 7645x3 − 18364x2 + 29429x − 23428. So π(G40 , x) ̸= π (G44 , x), which means that G19 and G65 are not per-cospectral. Similarly, we get that b5 (G13 )− b5 (G38 ) = −36, b5 (G39 )− b5 (G60 ) = 12, b5 (G22 )− b5 (G52 ) = −10 and b5 (G25 )− b5 (G62 ) = −12. These imply that pairs G13 and G38 , G39 and G60 , G22 and G52 , or G25 and G62 cannot be per-cospectral. By (v) of Lemma 2.1 and Table 2, we obtain that b4 (G13 ) − b4 (G18 ) = −4n + 32, b4 (G18 ) − b4 (G38 ) = 4n − 32. Hence b4 (G13 ) − b4 (G18 ) = 0 and b4 (G18 ) − b4 (G38 ) = 0 only when n = 8. However, by the definition of G13 , it can be seen that |V (G13 )| ≥ 9. This means that G13 and G18 cannot be per-cospectral. In addition, using Maple 12.0 with n = 8, we found that π (G18 , x) = x8 + 22x6 − 52x5 + 285x4 − 748x3 + 1720x2 − 2320x + 1412 and π (G38 , x) = x8 + 22x6 − 52x5 + 285x4 − 748x3 + 1756x2 − 2640x + 2200. These results indicate that G38 and G18 are not per-cospectral. Analogously, we have b4 (G18 ) − b4 (G25 ) = 4n − 24 and b4 (G18 ) − b4 (G62 ) = 4n − 24. Hence b4 (G18 ) − b4 (G25 ) = 0 and b4 (G18 ) − b4 (G62 ) = 0 only when n = 6. By the definitions of G25 and G62 , it can be seen that |V (G25 )| ≥ 8 and |V (G62 )| ≥ 7. These mean that neither pair G18 and G25 nor G18 and G62 are per-cospectral. Further we have b4 (G18 ) − b4 (G39 ) = 4n − 20 and b4 (G18 ) − b4 (G60 ) = 4n − 20. It is obvious that b4 (G18 ) − b4 (G39 ) = 0 and b4 (G18 ) − b4 (G60 ) = 0 only when n = 5. By the definitions of G39 and G60 , we have |V (G39 )| ≥ 6 and |V (G60 )| ≥ 7. These imply that neither pair G18 and G39 nor G18 and G62 are per-cospectral. Finally, by (v) of Lemma 2.1 and Table 2, we have b4 (G13 ) − b4 (G19 ) = 10, b4 (G13 ) − b4 (G22 ) = −2n + 21, b4 (G13 ) − b4 (G25 ) = 8, b4 (G13 )− b4 (G39 ) = 12, b4 (G13 )− b4 (G52 ) = −2n + 21, b4 (G13 )− b4 (G60 ) = 12, b4 (G13 )− b4 (G62 ) = 8, b4 (G13 )− b4 (G65 ) = 10, b4 (G18 ) − b4 (G19 ) = 4n − 22, b4 (G18 ) − b4 (G22 ) = 2n − 11, b4 (G18 ) − b4 (G52 ) = 2n − 11, b4 (G18 ) − b4 (G65 ) = 4n − 22, b4 (G19 )− b4 (G22 ) = −2n + 11, b4 (G19 )− b4 (G25 ) = −2, b4 (G19 )− b4 (G38 ) = −10, b4 (G19 )− b4 (G39 ) = 2, b4 (G19 )− b4 (G52 ) = −2n + 11, b4 (G19 ) − b4 (G60 ) = 2, b4 (G19 ) − b4 (G62 ) = −2, b4 (G22 ) − b4 (G25 ) = 2n − 13, b4 (G22 ) − b4 (G38 ) = 2n − 21, b4 (G22 ) − b4 (G39 ) = 2n − 9, b4 (G22 ) − b4 (G60 ) = 2n − 9, b4 (G22 ) − b4 (G62 ) = 2n − 13, b4 (G22 ) − b4 (G65 ) = 2n − 11, b4 (G25 )− b4 (G38 ) = −8, b4 (G25 )− b4 (G39 ) = 4, b4 (G25 )− b4 (G52 ) = −2n + 13, b4 (G25 )− b4 (G60 ) = 4, b4 (G25 )− b4 (G65 ) = 2, b4 (G38 )−b4 (G39 ) = 12, b4 (G38 )−b4 (G52 ) = −2n+21, b4 (G38 )−b4 (G60 ) = 12, b4 (G38 )−b4 (G62 ) = 8, b4 (G38 )−b4 (G65 ) = 10, b4 (G39 ) − b4 (G52 ) = −2n + 9, b4 (G39 ) − b4 (G62 ) = −4, b4 (G39 ) − b4 (G65 ) = −2, b4 (G52 ) − b4 (G60 ) = 2n − 9, b4 (G52 ) − b4 (G62 ) = 2n − 13, b4 (G52 )− b4 (G65 ) = 2n − 11, b4 (G60 )− b4 (G62 ) = −4, b4 (G60 )− b4 (G65 ) = −2 and b4 (G62 )− b4 (G65 ) = 2. We conclude that no pairs of G13 , G18 , G19 , G22 , G25 , G38 , G39 , G52 , G60 , G62 and G65 are per-cospectral.  4. Adjacency spectral characterization of every graph in Gn In this section, we will inquire into all graphs G in Gn which is DAS. By examining Fig. 1, we observe that each graph in Gn has the number of vertices at least 4. There exists only one graph when n = 4, i.e., G48 ∼ = 4K1 . It is obvious that G48 ∼ = 4K1 is DAS. Moreover, it can be seen that there exist five graphs with n = 5. By simple computation, we immediately obtain the following result. Lemma 4.1. In the five graphs of G5 , G41 , G46 and G47 are DAS, and G48 is cospectral with G49 .

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Lemma 4.2 ([4]). Graph Kn − E (Km ) (m ≤ n − 2) is DAS. By Lemma 4.2, we have Corollary 4.3. If n ≥ 6, then G48 = Kn − E (K4 ) is DAS. Lemma 4.4 ([4]). Kn − E (C6 ) and Kn − E (T2,2,2 ) with n ≥ 7 are cospectral, where T2,2,2 is obtained from K1,3 by attaching one edge on every pendant vertex. Suppose G25 and G62 have n vertices and m edges. We have the following result. Lemma 4.5. Let r (A(G)) denote the rank of A(G). Then r (A(G25 )) ≤ n − 1 and r (A(G62 )) = n. Proof. The adjacency matrices of G25 and G62 are expressed as

     A(G25 ) =    

0 0 1 1 1 1 1 1 1 1

0 0 0 1 0 1 1 1 1 1

1 0 0 0 1 0 1 1 1 1

1 1 0 0 1 1 1 1 1 1

1 0 1 1 0 1 1 1 1 1

1 1 0 1 1 0 1 1 1 1

1 1 1 1 1 1 0 0 1 1

1 1 1 1 1 1 0 0 1 1

1 1 1 1 1 1 1 1 0 1

1 1 1 1 1 1 1 1 1 0

··· ··· ··· ··· ··· ··· ··· ··· ··· ···

1 1 1 1 1 1 1 1 1 1

.. .. .. .. .. .. .. .. .. . . .. .. ........ .

     ,   

    A(G62 ) =    

1 1 1 1 1 1 1 1 1 1 ··· 0

0 0 1 1 1 1 1 1 1

0 0 0 1 1 1 0 1 1

1 0 0 0 1 1 1 1 1

1 1 0 0 0 1 1 1 1

1 1 1 0 0 0 1 1 1

1 1 1 1 0 0 1 1 1

1 0 1 1 1 1 0 1 1

1 1 1 1 1 1 1 0 1

1 1 1 1 1 1 1 1 0

··· ··· ··· ··· ··· ··· ··· ··· ···

1 1 1 1 1 1 1 1 1

.. .. .. .. .. .. .. .. . . .. .. ....... .

    .   

1 1 1 1 1 1 1 1 1 ··· 0

By checking A(G25 ), we find that the seventh and eighth rows are equal. So r (G25 ) ≤ n − 1. We can obtain that det(G62 ) = (−1)n−7 2(n − 6) ̸= 0. So r (G62 ) = n.  The corollary below follows easily from Lemma 4.5. Corollary 4.6. Graphs G25 and G62 are not cospectral. By Corollary 2.9 and (ii) of Lemma 2.8, we obtain the following result which can be thought of parallel result with Lemma 3.2 about adjacency spectrum. Lemma 4.7. Each of the following holds. (i) Graphs G3 , G5 and G6 are not pairwise cospectral. (ii) Graphs G55 and G66 are not cospectral. (iii) Graphs G4 , G7 , G8 , G10 and G14 are not pairwise cospectral. (iv) Graphs G43 , G46 , G54 , G57 and G67 are not pairwise cospectral. (v) Graphs G9 , G11 , G12 , G15 , G23 , G31 , G34 , G36 and G37 are not pairwise cospectral. (vi) Graphs G20 , G27 , G30 , G40 , G44 , G51 , G53 , G61 and G64 are not pairwise cospectral. (vii) Graphs G16 , G21 , G24 , G26 , G28 , G29 , G32 , G33 , G35 and G63 are not pairwise cospectral. (viii) Graphs G41 , G42 , G45 , G47 , G48 , G49 , G50 , G56 , G59 and G68 are not pairwise cospectral. (ix) Graphs G13 , G18 , G19 , G22 , G25 , G38 , G39 (or G60 ), G52 , G62 and G65 are not pairwise cospectral. Proof. By examining Fig. 1 and Lemma 4.1, we only consider n ≥ 6 for each graph in Gn . By Lemma 2.2, if two graphs do not have the same number of vertices, edges or closed walks of any fixed length, then they cannot be cospectral. In the proof of each case below, we always assume that the considered graphs have the same numbers of vertices and edges. Thus we can perform algebraic manipulations when applying (ii) of Lemma 2.8, Corollary 2.9, Table 2 and the corresponding results of b4 (Gi ) − b4 (Gj ) in the proof of Lemma 3.2 to find contradictions. (i) By Corollary 2.9 and Table 2, we have NG3 (4) − NG5 (4) = 16(c4 (G3 ) − c4 (G5 )) + 4(b4 (G5 ) − b4 (G3 )) = −8n + 36 ̸= 0. Hence by (ix) of Lemma 2.2, G3 and G5 are not cospectral. Similarly, by Corollary 2.9 and Table 2, we also have NG3 (4) − NG6 (4) = 8 and NG5 (4) − NG6 (4) = 8n − 28, which imply that G6 is not cospectral with either G3 or G5 . (ii) By Corollary 2.9 and Table 2, we have NG55 (4) − NG66 (4) = 8n − 52 ̸= 0. Lemma 2.2(ix) implies that G55 and G66 are not cospectral. (iii) By Lemmas 2.7, 2.8 and 2.10 and Table 3, we have NG4 (5)− NG7 (5) = 10[(NG4 (C5 )− NG7 (C5 ))+(D(G4 )− D(G7 ))] = 30. By (ix) of Lemma 2.2, G4 and G7 are not cospectral. By Corollary 2.9 and Table 2, we have NG4 (4) − NG8 (4) = −8n + 36, NG4 (4) − NG10 (4) = 8, NG4 (4) − NG14 (4) = 16, NG7 (4)− NG8 (4) = −8n + 36, NG7 (4)− NG10 (4) = 8, NG7 (4)− NG14 (4) = 16, NG8 (4)− NG10 (4) = 8n − 28, NG8 (4)− NG14 (4) = 8n − 20 and NG10 (4) − NG14 (4) = 8. By Lemma 2.2, we conclude that any two graphs in {G4 , G7 , G8 , G10 , G14 } are not cospectral. (iv) By Corollary 2.9 and Table 2, we have NG46 (4) − NG67 (4) = 16n − 96. Hence NG46 (4) − NG67 (4) = 0 only when n = 6. However, by definition of G67 , we found that |V (G67 )| ≥ 7. This means, by Lemma 2.2, that G46 and G67 cannot be cospectral. Similarly, by Corollary 2.9 and Table 2, we have NG43 (4) − NG46 (4) = −16n + 88, NG43 (4) − NG54 (4) = −8n + 44, NG43 (4) − NG57 (4) = 16, NG43 (4) − NG67 (4) = −8, NG46 (4) − NG54 (4) = 8n − 44, NG46 (4) − NG57 (4) = 16n − 72,

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NG54 (4) − NG57 (4) = 8n − 28, NG54 (4) − NG67 (4) = 8n − 52 and NG57 (4) − NG67 (4) = −24. It follows, by similar arguments and Lemma 2.2, that G43 , G46 , G54 , G57 and G67 are not pairwise cospectral. (v) By Lemmas 2.7, 2.8 and 2.10 and Table 3, we have NG11 (5)− NG23 (5) = −10n + 40. We note that NG11 (5)− NG23 (5) = 0 if and only if n = 4. This contradicts n ≥ 6. Hence by Lemma 2.2, G11 and G23 are not cospectral. Using the same argument as above, we also have NG15 (5) − NG31 (5) = 20, NG15 (5) − NG36 (5) = 30 and NG31 (5) − NG36 (5) = 10, which imply that none of pairs G15 and G31 , G15 and G36 , and G31 and G36 are cospectral. By Corollary 2.9 and Table 2, we obtain that NG11 (4) − NG37 (4) = −16n + 48 and NG23 (4) − NG37 (4) = −16n + 48, respectively. It is easy to see that NG11 (4) − NG37 (4) = 0 and NG23 (4) − NG37 (4) = 0 only when n = 3. These contradict n ≥ 6. By Lemma 2.2, we deduce that neither pair G11 and G37 nor G23 and G31 is cospectral. Similarly, by Corollary 2.9 and Table 2, we obtain that NG9 (4) − NG11 (4) = 24, NG9 (4) − NG12 (4) = −8n + 52, NG9 (4)− NG15 (4) = 16, NG9 (4)− NG23 (4) = 24, NG9 (4)− NG31 (4) = 16, NG9 (4)− NG34 (4) = −8n + 44, NG9 (4)− NG36 (4) = 16, NG9 (4) − NG37 (4) = −16n + 72, NG11 (4) − NG12 (4) = −8n + 28, NG11 (4) − NG15 (4) = −8, NG11 (4) − NG31 (4) = −8, NG11 (4) − NG34 (4) = −8n + 20, NG11 (4) − NG36 (4) = −8, NG12 (4) − NG15 (4) = 8n − 36, NG12 (4) − NG23 (4) = 8n − 28, NG12 (4) − NG31 (4) = 8n − 36, NG12 (4) − NG34 (4) = −8, NG12 (4) − NG36 (4) = 8n − 36, NG12 (4) − NG37 (4) = −8n + 20, NG15 (4) − NG23 (4) = 8, NG15 (4) − NG34 (4) = −8n + 28, NG15 (4) − NG37 (4) = −16n + 56, NG23 (4) − NG31 (4) = −8, NG23 (4) − NG34 (4) = −8n + 20, NG23 (4) − NG36 (4) = −8, NG31 (4) − NG34 (4) = −8n + 28, NG31 (4) − NG37 (4) = −16n + 56, NG34 (4) − NG36 (4) = 8n − 28, NG34 (4) − NG37 (4) = −8n + 28 and NG36 (4) − NG37 (4) = −16n + 56. By Lemma 2.2, we conclude that no pairs of G9 , G11 , G12 , G15 , G23 , G31 , G34 , G36 and G37 are cospectral. (vi) By Lemmas 2.7, 2.8 and 2.10 and Table 3, we get that NG40 (5)−NG44 (5) = 10n−40. It is clear that NG40 (5)−NG44 (5) = 0 only when n = 4. This contradicts n ≥ 6. It implies, by Lemma 2.2, that G40 and G44 are not cospectral. By Corollary 2.9 and Table 2, we have NG20 (4) − NG27 (4) = 8n − 44, NG20 (4) − NG30 (4) = 8n − 68, NG20 (4) − NG40 (4) = 8n − 12, NG20 (4) − NG44 (4) = 8n − 12, NG20 (4) − NG51 (4) = 24, NG20 (4) − NG53 (4) = 16, NG20 (4) − NG61 (4) = 8n − 20, NG20 (4)− NG64 (4) = 8n − 36, NG27 (4)− NG30 (4) = −24, NG27 (4)− NG40 (4) = 32, NG27 (4)− NG44 (4) = 32, NG27 (4)− NG51 (4) = −8n + 68, NG27 (4) − NG53 (4) = −8n + 60, NG27 (4) − NG61 (4) = 24, NG27 (4) − NG64 (4) = 8, NG30 (4) − NG40 (4) = 56, NG30 (4) − NG44 (4) = 56, NG30 (4) − NG51 (4) = −8n + 92, NG30 (4) − NG53 (4) = −8n + 84, NG30 (4) − NG61 (4) = 48, NG30 (4) − NG64 (4) = 32, NG40 (4) − NG51 (4) = −8n + 36, NG40 (4) − NG53 (4) = −8n + 28, NG40 (4) − NG61 (4) = −8, NG40 (4) − NG64 (4) = −24, NG44 (4) − NG51 (4) = −8n + 36, NG44 (4) − NG53 (4) = −8n + 28, NG44 (4) − NG61 (4) = −8, NG44 (4) − NG64 (4) = −24, NG51 (4) − NG53 (4) = −8, NG51 (4) − NG61 (4) = 8n − 44, NG51 (4) − NG64 (4) = 8n − 60, NG53 (4) − NG61 (4) = 8n − 36, NG53 (4) − NG64 (4) = 8n − 52 and NG61 (4) − NG64 (4) = −16. By Lemma 2.2, we deduce that no pairs of G20 , G27 , G30 , G40 , G44 , G51 , G53 , G61 and G64 are cospectral. (vii) Also we have NG24 (5) − NG28 (5) = 10n − 10 ̸= 0 when n ≥ 6. It implies that G24 and G28 are not cospectral by Lemma 2.2. Similarly, we have NG26 (5) − NG63 (5) = 10, and G26 is not cospectral with G63 . By Corollary 2.9 and Table 2, we have NG16 (4) − NG21 (4) = −8n + 28, NG16 (4) − NG24 (4) = −16, NG16 (4) − NG26 (4) = −8, NG16 (4) − NG28 (4) = −16, NG16 (4) − NG29 (4) = −8n + 12, NG16 (4) − NG32 (4) = −24, NG16 (4) − NG33 (4) = −8n − 4, NG16 (4) − NG35 (4) = −32, NG16 (4) − NG63 (4) = −8, NG21 (4) − NG24 (4) = 8n − 44, NG21 (4) − NG26 (4) = 8n − 36, NG21 (4) − NG28 (4) = 8n − 44, NG21 (4) − NG29 (4) = −16, NG21 (4) − NG32 (4) = 8n − 52, NG21 (4) − NG33 (4) = −32, NG21 (4) − NG35 (4) = 8n − 60, NG21 (4) − NG63 (4) = 8n − 36, NG24 (4) − NG26 (4) = 8, NG24 (4) − NG29 (4) = −8n + 28, NG24 (4)−NG32 (4) = −8, NG24 (4)−NG33 (4) = −8n+12, NG24 (4)−NG35 (4) = −16, NG24 (4)−NG63 (4) = 8, NG26 (4)−NG28 (4) = −8, NG26 (4) − NG29 (4) = −8n + 20, NG26 (4) − NG32 (4) = −16, NG26 (4) − NG33 (4) = −8n + 4, NG26 (4) − NG35 (4) = −24, NG28 (4) − NG29 (4) = −8n + 28, NG28 (4) − NG32 (4) = −8, NG28 (4) − NG33 (4) = −8n + 12, NG28 (4) − NG35 (4) = −16, NG28 (4) − NG63 (4) = 8, NG29 (4) − NG32 (4) = 8n − 36, NG29 (4) − NG33 (4) = −16, NG29 (4) − NG35 (4) = 8n − 44, NG29 (4) − NG63 (4) = 8n − 20, NG32 (4) − NG33 (4) = −8n + 20, NG32 (4) − NG35 (4) = −8, NG32 (4) − NG63 (4) = 16, NG33 (4) − NG35 (4) = 8n − 28, NG33 (4) − NG63 (4) = 8n − 4 and NG35 (4) − NG63 (4) = 24. By Lemma 2.2, these results indicate that G16 , G21 , G24 , G26 , G28 , G29 , G32 , G33 , G35 and G63 are not pairwise cospectral. (viii) Similarly we have NG41 (5) − NG50 (5) = −10n + 50 ̸= 0 when n ≥ 6. It means that G41 and G50 are not cospectral by Lemma 2.2. By Corollary 2.9 and Table 2, we have NG47 (4) − NG59 (4) = 16n − 112. Hence NG47 (4) − NG59 (4) = 0 only when n = 7. Employing Maple 12.0 with n = 7. We discovered that φ(G47 , x) = x7 + 15x5 − 26x4 + 102x3 − 160x2 + 178x − 72 and φ(G59 , x) = x7 + 15x5 − 26x4 + 106x3 − 174x2 + 222x − 134. These results indicate, by Lemma 2.2, that G47 is not cospectral with G59 . By the similar argument as above, we have NG42 (4) − NG47 (4) = −16n + 80 ̸= 0 and NG45 (4) − NG49 (4) = −16n + 64 ̸= 0 when n ≥ 6. They imply that neither pair G42 and G47 nor G45 and G49 is cospectral. Similarly, by Corollary 2.9 and Table 2, we also have NG47 (4) − NG68 (4) = 16n − 96. Hence NG45 (4) − NG49 (4) = 0 only when n = 6. However, by the definition of G68 , we note that |V (G68 )| ≥ 7. This means that G47 is not cospectral with G68 . Finally, by Corollary 2.9 and Table 2, we have NG41 (4) − NG42 (4) = 8n − 44, NG41 (4) − NG45 (4) = 8n − 52, NG41 (4) − NG47 (4) = −8n + 36, NG41 (4) − NG49 (4) = −8n + 12, NG41 (4) − NG56 (4) = −16, NG41 (4) − NG59 (4) = 8n − 76, NG41 (4) − NG68 (4) = 8n − 60, NG42 (4) − NG45 (4) = −8, NG42 (4) − NG49 (4) = −16n + 56, NG42 (4) − NG50 (4) = −8n + 44, NG42 (4) − NG56 (4) = −8n + 28, NG42 (4) − NG59 (4) = −32, NG42 (4) − NG68 (4) = −16, NG45 (4) − NG47 (4) = −16n + 88, NG45 (4) − NG50 (4) = −8n + 52, NG45 (4) − NG56 (4) = −8n + 36, NG45 (4) − NG59 (4) = −24, NG45 (4) − NG68 (4) = −8, NG47 (4) − NG49 (4) = −24, NG47 (4) − NG50 (4) = 8n − 36, NG47 (4) − NG56 (4) = 8n − 52, NG49 (4) − NG50 (4) = 8n − 12, NG49 (4) − NG56 (4) = 8n − 28, NG49 (4) − NG59 (4) = 16n − 88, NG49 (4) − NG68 (4) = 16n − 72, NG50 (4) − NG56 (4) = −16, NG50 (4) − NG59 (4) = 8n − 76, NG50 (4) − NG68 (4) = 8n − 60, NG56 (4) − NG59 (4) = 8n − 60, NG56 (4) − NG68 (4) = 8n − 44

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and NG59 (4) − NG68 (4) = 16. By Lemma 2.2, we conclude that no pairs of G41 , G42 , G45 , G47 , G49 , G50 , G56 , G59 and G68 are cospectral. (ix) By Lemma 4.4, we observe that G39 and G60 are cospectral when n ≥ 7. Without loss of generality, we choose G39 as representative in this case. By Lemmas 2.7, 2.8 and 2.10 and Table 3, we have NG13 (5) − NG38 (5) = 60 and NG22 (5) − NG52 (5) = 10. These imply, by Lemma 2.2, that neither pair G13 and G38 nor G22 and G52 is cospectral. Similarly, we also have NG19 (5)− NG65 (5) = −10n + 50. It is easy to see that NG19 (5) − NG65 (5) = 0 if and only if n = 5. This is a contradiction since n ≥ 6. Thus G19 and G65 cannot be cospectral. By Corollary 2.9 and Table 2, we obtain NG13 (4)− NG18 (4) = −16n + 112 and NG18 (4)− NG38 (4) = 16n − 112, respectively. Hence NG13 (4)− NG18 (4) = 0 and NG18 (4)− NG38 (4) = 0 only when n = 7. However, by the definitions of G13 and G38 , we note that |V (G13 )| ≥ 9 and |V (G13 )| ≥ 8. By Lemma 2.2, these mean that neither pair G13 and G38 nor G18 and G38 is cospectral. Similarly, we also obtain that NG18 (4)− NG25 (4) = 16n − 80, NG18 (4)− NG62 (4) = 16n − 80 and NG18 (4)− NG39 (4) = 16n − 64. It is obvious that NG18 (4) − NG25 (4) = 0, NG18 (4) − NG62 (4) = 0 and NG18 (4) − NG39 (4) = 0 only when n = 5, n = 5 and n = 4. These contradict n ≥ 6. So, none of pairs G18 and G25 , G18 and G62 , and G18 and G39 are cospectral. Finally, by Corollary 2.9 and Table 2, we have NG13 (4)− NG19 (4) = 40, NG13 (4)− NG22 (4) = −8n + 76, NG13 (4)− NG25 (4) = 32, NG13 (4) − NG39 (4) = 48, NG13 (4) − NG52 (4) = −8n + 76, NG13 (4) − NG62 (4) = 32, NG13 (4) − NG65 (4) = 40, NG18 (4) − NG19 (4) = 16n − 72, NG18 (4) − NG22 (4) = 8n − 36, NG18 (4) − NG52 (4) = 8n − 36, NG18 (4) − NG65 (4) = 16n − 72, NG19 (4)−NG22 (4) = −8n+36, NG19 (4)−NG25 (4) = −8, NG19 (4)−NG38 (4) = −40, NG19 (4)−NG39 (4) = 8, NG19 (4)−NG52 (4) = −8n + 36, NG19 (4) − NG62 (4) = −8, NG22 (4) − NG25 (4) = 8n − 44, NG22 (4) − NG38 (4) = 8n − 76, NG22 (4) − NG39 (4) = 8n − 28, NG22 (4) − NG62 (4) = 8n − 44, NG22 (4) − NG65 (4) = 8n − 36, NG25 (4) − NG38 (4) = −32, NG25 (4) − NG39 (4) = 16, NG25 (4) − NG52 (4) = −8n + 44, NG25 (4) − NG65 (4) = 8, NG38 (4) − NG39 (4) = 48, NG38 (4) − NG52 (4) = −8n + 76, NG38 (4) − NG62 (4) = 32, NG38 (4) − NG65 (4) = 40, NG39 (4) − NG52 (4) = −8n + 28, NG39 (4) − NG62 (4) = −16, NG39 (4) − NG65 (4) = −8, NG52 (4) − NG62 (4) = 8n − 44, NG52 (4) − NG65 (4) = 8n − 36 and NG62 (4) − NG65 (4) = 8. These imply, by Lemma 2.2, that no pairs of G13 , G18 , G19 , G22 , G25 , G38 , G39 (or G60 ), G52 , G62 and G65 are cospectral.  Summing up Corollaries 4.3 and 4.6 and Lemmas 2.4, 4.1, 4.4 and 4.7, we have the following main result. Theorem 4.8. Let G be a graph in Gn . (i) If n = 4 and 6, then G is DAS. (ii) If n = 5, then G is DAS if and only if G ̸∈ {K5 − E (K4 ), K5 − E (B)}, where B denotes a bowtie. (iii) If n ≥ 7, then G is DAS if and only if G ̸∈ {Kn − E (C6 ), Kn − E (T2,2,2 )}. 5. Discussions As a natural tool of the study of graphs the permanental polynomial of adjacency matrix of a graph was introduced independently in mathematics [23,29] and in chemistry [17,28]. The coefficient of such polynomial reveals many structural properties of graphs; for example, see Lemmas 2.1 and 2.2. In particular, the absolute value of its constant expressing as the permanent of the adjacency matrix can be used to count perfect matchings of bipartite graphs, which coincide with Kekulé structures of alternant benzenoids and dimers of square lattices respectively [22,26]. However, it is hard to compute the permanent and permanental polynomial. Recently some algorithms were developed for computing the permanental polynomial of some fullerenes [5,6,8,18]. Such computation results show that both permanental polynomials of IPR C60 and C70 have some zero properties consistent with the data of C≤50 . That is, there are n/2 independent zeros for each permanental polynomial, ten of them are nearly a constant for all fullerene isomers with fixed number n of vertices, while the remaining n/2 − 10 zeros vary with structure. In 1968 Turner [29] introduced so-called immanantal polynomials of a graph including characteristic polynomial and permanental polynomial as its special cases to characterize graphs. Unfortunately, examples were given of nonisomorphic graphs that have the same immanantal polynomial. Merris et al. [23] first observed that the five pairs of cospectral graphs (see [16]) are distinguished by their permanental polynomials. Moreover, they stated that the per-spectrum seems a little better than the adjacency spectrum when it comes to distinguishing graphs which are not trees. In this paper, we first showed that all graphs obtained by Kn deleting six edges are DPS; see Theorem 3.1. Then, using the relation of the 4th coefficient of permanental polynomials of these graphs and their the number of walks of length 4, we showed that there exist just two pairs of cospectral graphs in these graphs; see Theorem 4.8. When at most five edges are deleted from Kn , Cámara and Haemers [4] showed that there is just one pair of nonisomorphic cospectral graphs. But these graphs are all DPS [36]. Further recent work [33] showed that complete bipartite graphs are all DPS; However, it has been not unsolved yet to determine whether they are DAS. Such results as above provide affirmative support for Merris’ statement. Nevertheless, the considered graphs are dense and the present results cannot be directly applied to chemical molecular graphs. It is interesting to compare the effects of distinguishing chemical molecular graphs with permanental polynomial and characteristic polynomials. Further problems are to search for pairs of per-cospectral benzenoids and fullerenes respectively, and to characterize which chemical molecular graphs are DPS.

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Acknowledgment The authors would like to thank the anonymous referee for giving valuable suggestions and comments to improve the manuscript. References [1] F. Belardo, V. De Filippis, S.K. Simić, Computing the permanental polynomial of a matrix from a combinatorial viewpoint, MATCH Commun. Math. Comput. Chem. 66 (2011) 381–396. [2] M. Borowiecki, On spectrum and per-spectrum of graphs, Publ. Inst. Math. (Beograd) 38 (1985) 31–33. [3] R. Boulet, B. Jouve, The lollipop graphs is determined by its spectrum, Electron. J. Combin. 15 (2008) #R74. [4] M. Cámara, W.H. Haemers, Spectral characterizations of almost complete graphs, Discrete Appl. Math. 176 (2014) 19–23. [5] G.G. Cash, The permanental polynomial, J. Chem. Inf. Comput. Sci. 40 (2000) 1203–1206. [6] G.G. Cash, Permanental polynomials of smaller fullerenes, J. Chem. Inf. Comput. Sci. 40 (2000) 1207–1209. [7] R. Chen, A note on the relations between the permanental and characteristic polynomials of coronoid hydrocarbons, MATCH Commun. Math. Comput. Chem. 51 (2004) 137–148. [8] Q. Chou, H. Liang, F. Bai, Computing the permanental polynomial of the high level fullerene C70 with high precision, MATCH Commun. Math. Comput. Chem. 73 (2015) 327–336. [9] Q. Chou, H. Liang, F. Bai, Remarks on the relations between the permanental and characteristic polynomials of fullerenes, MATCH Commun. Math. Comput. Chem. 66 (2011) 743–750. [10] D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs, Academic Press, New York, 1980. [11] M. Doob, W.H. Haemers, The complement of the path is determined by its spectrum, Linear Algebra Appl. 356 (2002) 57–65.

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