Linear Algebra and its Applications 452 (2014) 281–291
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Linear Algebra and its Applications www.elsevier.com/locate/laa
Three distance characteristic polynomials of some graphs ✩ Changxiang He ∗ , Shiqiong Liu, Baofeng Wu College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
a r t i c l e
i n f o
Article history: Received 25 September 2013 Accepted 28 March 2014 Available online xxxx Submitted by R. Brualdi MSC: 05C50 05C12 Keywords: Distance eigenvalue Equitable partition Graph join
a b s t r a c t We give complete information about the distance, distance Laplacian and distance signless Laplacian characteristic polynomials of graphs obtained by a generalized join graph operation on families of graphs. As an application of these results, we construct many pairs of nonisomorphic distance, distance Laplacian and distance signless Laplacian cospectral graphs, and then give a negative answer to the question “Can every connected graph be determined by its distance Laplacian spectrum and/or distance signless Laplacian spectrum?” proposed in Aouchiche and Hansen (2013) [2]. © 2014 Elsevier Inc. All rights reserved.
1. Introduction The distance matrix is more complex than the ordinary adjacency matrix of a graph since the distance matrix is a complete matrix (dense) while the adjacency matrix often is very sparse. Thus the computation of the characteristic polynomial of the distance matrix is computationally a much more intense problem and, in general, there are no ✩ Research supported by the Natural Science Foundation of Shanghai (Grant No. 12ZR1420300), National Natural Science Foundation of China (Nos. 11101284, 11201303 and 11301340). * Corresponding author. E-mail address:
[email protected] (C. He).
http://dx.doi.org/10.1016/j.laa.2014.03.045 0024-3795/© 2014 Elsevier Inc. All rights reserved.
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simple analytical solutions except for a few trees. The distance matrix of a graph has numerous applications to chemistry and other branches of science. The distance matrix, containing information on various walks and self-avoiding walks of chemical graphs, is immensely useful in the computation of topological indices such as the Wiener index, is useful in the computation of thermodynamic properties such as pressure and temperature coefficients and it contains more structural information compared to a simple adjacency matrix. For a survey see [1] and also the papers cited therein. All graphs considered here are simple and undirected. Let G = (V (G), E(G)) be a graph with vertex set V (G) = {1, 2, . . . , n}. The adjacency matrix of G, denoted by A(G), is the n × n matrix whose (i, j)-entry is 1 if i and j are adjacent and 0 otherwise. Let D(G) be the diagonal degree matrix of G, where the (i, i)-entry is equal to dG (i), the degree of vertex i. Then the Laplacian matrix of G is L(G) = D(G) − A(G) and the signless Laplacian matrix of G is Q(G) = D(G) + A(G). For i, j ∈ V (G), the distance between i and j, denoted by dG (i, j), is the length of a shortest path from i to j in G. The distance matrix of a connected graph G is the n × n matrix D(G) = (dG (i, j)). The transmission Tr(i) of a vertex i is the sum of the n distances from i to all other vertices, i.e., Tr(i) = j=1 dG (i, j). A connected graph is said to be distance regular if the transmission is a constant for every vertex. Let T (G) be the diagonal transmission matrix of G, where the (i, i)-entry is equal to Tr(i). Similarly to the Laplacian and signless Laplacian, Aouchiche and Hansen [2] defined the distance Laplacian and distance signless Laplacian of a connected graph G as the matrices DL (G) = T (G) − D(G) and DQ (G) = T (G) + D(G), respectively. For a graph G, as we see, there are many matrices associated with G. Let M = M (G) be a matrix associated with G. The M -polynomial is defined as ΦM (G, x) = det(xI −M ), where I is the identity matrix. The M -eigenvalues are the roots of the M -polynomial, and the M -spectrum of G is a multiset consisting of the M -eigenvalues. A graph is called M -integral if its M -spectrum consists only of integers. Graphs with the same M -spectrum are called M -cospectral graphs. Two M -cospectral non-isomorphic graphs G and H are called M -cospectral mates or M -mates. Aouchiche and Hansen [2] propose the following question. Question 1.1. Can every connected graph be determined by its DL -spectrum and/or DQ -spectrum? For two disjoint graphs G and H, let G ∪ H denote the union of G and H. And let G∨H be the graph obtained from G∪H by joining every vertex of G to every vertex of H. The union and join may be viewed as special cases of a more general operation which are called “generalization composition” in [3]. If G is labeled and has k vertices, then the graph G[H1 , . . . , Hk ] is formed by taking the disjoint graphs H1 , . . . , Hk and then joining every vertex of Hi to every vertex of Hj when i is adjacent to j in G. Thus the join is given by G ∨ H = K2 [G, H]. Schwenk [3] and Cardoso et al. [4] provided complete information about the A-spectrum of G[H1 , . . . , Hk ] for any graph G and regular graphs H1 , . . . , Hk .
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Cardoso et al. [4] also gave complete information about the L-spectrum of G[H1 , . . . , Hk ] for any graph G and arbitrary graphs H1 , . . . , Hk . Motivated by these works, we give complete information about the D-polynomial, DQ -polynomial of G[H1 , . . . , Hk ] for any connected graph G and particular graphs H1 , . . . , Hk , and the DL -polynomial of G[H1 , . . . , Hk ] for any connected graph G and arbitrary graphs H1 , . . . , Hk . The rest of the paper is organized as follows. In Section 2, we introduce equitable partition of symmetric real matrix and state some related results on eigenvalues. Section 3 is dedicated to study the D-polynomial, DL -polynomial and DQ -polynomial of G[H1 , . . . , Hk ]. Finally, as an application of these results, we construct many pairs of nonisomorphic distance, distance Laplacian, distance signless Laplacian cospectral graphs and answer Question 1.1. 2. Preliminaries Denote the all-one vector of order n by 1n and the all-one matrix of order m × n by Jm×n . For all other notations and definitions not given here, the readers refereed to [5]. Definition 2.1. (See [5].) Suppose M is a symmetric real matrix whose rows and columns are indexed by X = {1, 2, . . . , n}. Let X1 , . . . , Xk be a k-partition (a partition with exactly k cells) of X, say π, denote the submatrix of M with row set Xi and column set Xj by Mij . The characteristic matrix Bπ is the n × k matrix whose j-th column is the characteristic vector of Xj (j = 1, . . . , k). If the row sum of each block Mij is constant, then the partition is called equitable. Let Mπ be the k × k matrix whose (i, j)-entry is the row sum of Mij , we call Mπ the divisor matrix of M with respect to π. The following proposition characterizes the equitable partition of a matrix. Proposition 2.1. Let M be a matrix, π be an equitable partition of M with divisor matrix Mπ and characteristic matrix Bπ , then x is an eigenvector of Mπ if and only if Bπ x is an eigenvector of M corresponding to the same eigenvalue. Proof. It was proved in [6] (see p. 24) that M Bπ = Bπ Mπ . So the necessity is clear. For the sufficiency, let Bπ x be an eigenvector of M corresponding to λ, then λBπ x = M Bπ x = Bπ Mπ x, it follows that Bπ (λx − Mπ x) = 0. Note that the columns of Bπ are linearly independent, so we can conclude that Mπ x = λx. 2 Proposition 2.2. Let M be a nonnegative irreducible matrix and π be an equitable partition of M with divisor matrix Mπ . Then Mπ has the largest eigenvalue of M as its eigenvalue. Proof. Suppose π has k cells with orders n1 , · · · , nk . Since M is a nonnegative irreducible matrix, thus Mπ is also nonnegative and irreducible. It follows from the famous Perron Theorem that there exists a positive eigenvector x = (x1 , x2 , . . . , xk )T corresponding to the largest eigenvalue of Mπ , say λ. Let Bπ be the characteristic matrix of π. Then
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⎡
x1 1n1 ⎢ ⎢ x2 1n2 y = Bπ x = ⎢ ⎢ .. ⎣ . xk 1nk
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
is an eigenvector of M corresponding to λ from Proposition 2.1. If w is an eigenvector of M orthogonal to y, in view of that x1 , · · · , xk are all positive, then w is neither a positive nor a negative vector. Hence, w is not an eigenvector corresponding to the largest eigenvalue of M , which implies that λ is the largest eigenvalue of M . 2 3. Distance eigenvalues of G[H1 , . . . , Hk ] In this section, we will study the three distance eigenvalues of some particular graphs. For convenience, for any graph G and constants a, b, we define Aa,b (G) = aD(G)+bA(G) and Da,b (G) = aT (G) + bD(G). For example, A0,1 (G) = A(G), A1,−1 (G) = L(G), A1,1 (G) = Q(G), D0,1 (G) = D(G), D1,−1 (G) = DL (G) and D1,1 (G) = DQ (G). If G is connected and a 0, b > 0, then Aa,b (G) and Da,b (G) are nonnegative and irreducible. We next focus on the eigenvalues of Da,b (G[H1 , . . . , Hk ]). For any connected graph G with V (G) = {1, 2, . . . , k}, considering the graph G[H1 , . . . , Hk ], where Hi (i = 1, · · · , k) is a graph of order ni , we define Ni = 2(ni − 1) +
k
nj dG (i, j),
i ∈ V (G).
j=1
The numbers Ni (i = 1, · · · , k) play important roles in the following, and they only dependent on the graph G and the orders, not the structures, of graphs H1 , . . . , Hk . 3.1. Distance characteristic polynomial Note that for a matrix M , the eigenvalue λ is a main eigenvalue if the eigenspace E(λ) is not orthogonal to the all-one vector 1. For A(G), it is well known that any regular graph G has exactly one main eigenvalue, and V (G) is an equitable 1-partition of A(G); any (s, t)-semiregular bipartite graph (a bipartite graph whose each vertex in the first (resp. √ second) color class has degree s (resp. t)) has exactly two main eigenvalues ± st, and obviously the two color sets is an equitable 2-partition of its adjacency matrix. For L(G), the Laplacian matrix of any connected graph has exactly one main eigenvalue 0, and V (G) is an equitable 1-partition of L(G). We next determine the characteristic polynomials of Da,b (G[H1 , . . . , Hk ]), where G is connected, Hi (i = 1, . . . , k) is a graph with adjacency matrix A(Hi ), and satisfies that Aa,b (Hi ) has an equitable ri -partition, where ri is the number of the main eigenvalues of Aa,b (Hi ).
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Theorem 3.1. Let G be a connected graph of order k 2, Hi (i = 1, . . . , k) be a graph of order ni . Let a and b be two constants, and all the main eigenvalues of the matrix Aa,b (Hi ) are λi1 , . . . , λiri . If Vi1 , . . . , Viri is an equitable ri -partition of Aa,b (Hi ), then V11 , . . . , V1r1 , . . . , Vk1 , . . . , Vkrk is an equitable partition of Da,b = Da,b (G[H1 , . . . , Hk ]). Denote the corresponding divisor matrix by Mπ , then Φ(Da,b , x) = Φ(Mπ , x)
n
(−1)ni −ri
i=1
Φ(aD(Hi ) + bA(Hi ), aNi − 2b − x) . (aNi − 2b − λi1 − x) · . . . · (aNi − 2b − λiri − x)
Proof. For short, set G = G[H1 , . . . , Hk ] and abbreviate dG (i, j) to dij . It is obvious that V11 , . . . , V1r1 , . . . , Vk1 , . . . , Vkrk is an equitable partition of Da,b (G ), denote the corresponding characteristic matrix by Bπ . By a suitable ordering of vertices in G , its distance matrix D(G ) can be written as ⎡
2Jn1 ×n1 − 2In1 − A(H1 ) d21 Jn2 ×n1 ⎢ D G = ⎢ .. ⎣ . dk1 Jnk ×n1
d12 Jn1 ×n2 2Jn2 ×n2 − 2In2 − A(H2 ) .. . dk2 Jnk ×n2
... ... .. . ...
⎤
d1k Jn1 ×nk d2k Jn2 ×nk ⎥ ⎥, .. ⎦ . 2Jnk ×nk − 2Ink − A(Hk )
then for any vertex v ∈ V (Hi ), k Tr G (v) = dHi (v) + 2 ni − 1 − dHi (v) + nj dij = Ni − dHi (v). j=1
Thus T (G ) = diag(N1 In1 − D(H1 ), N2 In2 − D(H2 ), · · · , Nk Ink − D(Hk )). So we have ⎡
M1 ⎢ bd ⎢ 21 Jn2 ×n1 Da,b G = ⎢ .. ⎢ . ⎣ bdk1 Jnk ×n1
bd12 Jn1 ×n2 M2 .. . bdk2 Jnk ×n2
... ... .. . ...
⎤ bd1k Jn1 ×nk ⎥ bd2k Jnk ×n2 ⎥ ⎥, .. ⎥ . ⎦ Mk
where Mi = 2bJni ×ni + (aNi − 2b)Ini − Aa,b (Hi ), i = 1, . . . , k. In the following we want to find out all the eigenvalues of Da,b (G ) by considering eigenvectors associated with them. Let βi = (z1 , z2 , . . . , zni )T be an eigenvector of Aa,b (Hi ) orthogonal to 1ni and satisfies Aa,b (Hi )β i = λβ i . Let z = (0, . . . , 0, z1 , z2 , . . . , zni , 0, . . . , 0)T , where zj (j = 1, . . . , ni ) corresponds to the vertices in Hi . Then Da,b (G )z = (aNi − 2b − λ)z, which implies that z is an eigenvector corresponding to the eigenvalue aNi − 2b − λ of Da,b (G ). Let ⎡
⎤ w1 ⎥ ⎢ w = ⎣ ... ⎦ wk
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be an eigenvector of Da,b (G ) orthogonal to z, where wi (i = 1, . . . , k) corresponds to the vertices in Hi . Then wi (i = 1, . . . , k) is orthogonal to β i , i.e., wi = xi 1ni , it is not difficult to see that ⎡
⎤ x1 1r1 ⎢ ⎥ w = Bπ ⎣ ... ⎦ . xn 1rk By Proposition 2.1, w is an eigenvector of Da,b (G ) corresponding to λ if and only if ⎡
⎤ x1 1r1 ⎢ .. ⎥ ⎣ . ⎦ xn 1rk is an eigenvector of Mπ corresponding to λ. This form a complete set of eigenvectors of Da,b (G ). For each graph Hi , we divide Φ(Da,b (G ), x) by (aNi − 2b − λi1 − x) · . . . · (aNi − 2b − λiri − x) to remove the factor corresponding to main eigenvalues of Aa,b (Hi ). Multiplying by Φ(Mπ , x) completes the proof. 2 In the above theorem, if we take a = 0, b = 1 and a = 1, b = 1, then we obtain the distance and distance signless Laplacian characteristic polynomial of G[H1 , . . . , Hk ], respectively. Corollary 3.1. Let G be a connected graph of order k 2, Hi (i = 1, . . . , k) be a graph of order ni with all main eigenvalues λi1 , . . . , λiri . If Vi1 , . . . , Viri is an equitable ri -partition of Hi , then V11 , . . . , V1r1 , . . . , Vk1 , . . . , Vkrk is an equitable partition of D(G[H1 , . . . , Hk ]). Denote the corresponding divisor matrix by Dπ , then k
ΦD G[H1 , . . . , Hk ], x = Φ(Dπ , x) (−1)ni i=1
ΦA (Hi , −x − 2) . (x + λi1 + 2) · . . . · (x + λiri + 2)
Corollary 3.2. Let G be a connected graph of order k 2, Hi (i = 1, . . . , k) be a graph of order ni . Let all the main eigenvalues of Q(Hi ) be qi1 , . . . , qiri . If Vi1 , . . . , Viri is an equitable ri -partition of Hi , then V11 , . . . , V1r1 , . . . , Vk1 , . . . , Vkrk is an equitable partition of DQ (G[H1 , . . . , Hk ]). Denote the corresponding divisor matrix by DπQ , then k
ΦDQ G[H1 , . . . , Hk ], x = Φ DπQ , x (−1)ni i=1
ΦQ (Hi , −x + Ni − 2) . (x + qi1 − Ni + 2) · . . . · (x + qiri − Ni + 2)
If we take a = 1, b = −1 in Theorem 3.1, then Da,b (G ) = DL (G[H1 , . . . , Hk ]). It is well known that if Hi has exactly ωi components Hi1 , Hi2 , . . . , Hiωi , then L(H) has
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exactly ωi main eigenvalues 0 and V (Hi1 ), . . . , V (Hiωi ) is an equitable ωi -partition of the matrix L(Hi ), so we have the following result. Corollary 3.3. Let G be a connected graph of order k 2, Hi (i = 1, . . . , k) be a graph of order ni and exactly ωi components Hi1 , Hi2 , . . . , Hiωi , then V (H11 ), . . . , V (H1ω1 ), . . . , V (Hk1 ), . . . , V (Hkωk ) is an equitable partition of the matrix DL (G[H1 , . . . , Hk ]). Denote the corresponding divisor matrix by DπL , then k
ΦL (Hi , −x + Ni + 2) ΦDL G[H1 , . . . , Hk ], x = Φ DπL , x (−1)ni . (x − Ni − 2)ωi i=1
Corollary 3.4. Let G be a connected graph of order k 2, Hi (i = 1, . . . , k) be a connected graph of order ni . Then k
ΦL (Hi , −x + Ni + 2) , ΦDL G[H1 , . . . , Hk ], x = Φ DπL , x (−1)ni x − Ni − 2 i=1
where DπL is the matrix with L Dπ ij =
−2(ni − 1) + Ni
if i = j
−nj dG (i, j)
if i = j.
3.2. Cospectrality If we assume that H1 , . . . , Hn are all regular graphs in Theorem 3.1, then V (H1 ), . . . , V (Hn ) is an equitable partition of D(G[H1 , . . . , Hk ]), we can deduce the following result. Corollary 3.5. Let G be a connected graph of order k 2, Hi (i = 1, . . . , k) be an ri -regular graph of order ni . Then k
ΦA (Hi , −x − 2) , ΦD G[H1 , . . . , Hk ], x = Φ(Dπ , x) (−1)ni x + ri + 2 i=1
(1)
where Dπ is the matrix with
(Dπ )ij =
and
2(ni − 1) − ri
if i = j
nj dG (i, j)
if i = j,
k
ΦQ (Hi , −x + Ni − 2) ΦDQ G[H1 , . . . , Hk ], x = Φ DπQ , x , (−1)ni x + 2ri − Ni + 2 i=1
(2)
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where DπQ is the matrix with
DπQ ij
=
Ni + 2(ni − 1) − 2ri nj dG (i, j)
if i = j if i = j.
The following is an immediate consequence of Corollary 3.5. Corollary 3.6. (1) If G1 and G2 are connected D-mates of order k 2, H1 , . . . , Hk are regular graphs with same order and degree, then G1 [H1 , . . . , Hk ] and G2 [H1 , . . . , Hk ] are D-mates. Especially, if G1 and G2 are distance regular D-mates, then G1 [H1 , . . . , Hk ] and G2 [H1 , . . . , Hk ] are DQ -mates. (2) If G is a connected graph of order k 2, Gi and Hi (i = 1, . . . , k) are regular A-mates, then G[G1 , . . . , Gk ] and G[H1 , . . . , Hk ] are D-mates and DQ -mates. Proof. (1) In Corollary 3.5, if we assume H1 , . . . , Hk are all r-regular graphs of order n, then Dπ = (2n − 2 − r)Ik + nD(G), so ΦD (G[H1 , . . . , Hk ], x) is determined by D(G) and A(Hi ). Based on this, we have the first result. If G1 and G2 are distance regular, it is easy to check that both G1 [H1 , . . . , Hk ] and G2 [H1 , . . . , Hk ] are distance regular, form the definition of distance signless Laplacian matrix, we know that they are also DQ -mates. (2) For fixed graph G in Corollary 3.5, Ni (i = 1, . . . , k), Dπ and DπQ are determined by the orders and degrees of regular graphs H1 , . . . , Hk . For any i ∈ {1, 2, . . . , k}, if Gi and Hi are regular A-mates, then they have the same order and degree. This lead us to the desired result. 2 Corollary 3.7. (1) If G1 and G2 are connected distance regular DL -mates of order k 2, H1 , . . . , Hk are connected graphs with same order, then G1 [H1 , . . . , Hk ] and G2 [H1 , . . . , Hk ] are DL -mates. (2) If G is a connected graph of order k 2, Gi and Hi (i = 1, . . . , k) are connected L-mates, then G[G1 , . . . , Gk ] and G[H1 , . . . , Hk ] are DL -mates. Proof. (1) In Corollary 3.4, if we assume the orders of H1 , . . . , Hk are all n, then DπL = nDL (G). So ΦDL (G[H1 , . . . , Hk ], x) is determined by DL (G), L(Hi ) and Ni . If G is distance regular, then Ni is a constant for each i ∈ {1, 2 . . . , k}. Based on this, we have the desired result. (2) For fixed graph G in Corollary 3.4, Ni (i = 1, . . . , k) and DπL are determined by the orders of H1 , . . . , Hk . For any i ∈ {1, 2 . . . , k}, if Gi and Hi are L-mates, then they have the same order. This together with Corollary 3.4, lead us to the desired result. 2
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Fig. 3.1. A pair of cospectral regular graphs.
Now we are ready to answer the following two questions proposed in [2]. Question 3.1. Can every connected graph be determined by its DL -spectrum and/or DQ -spectrum? Since there are many pairs of Laplacian cospectral connected graphs in [7–12], by Corollary 3.7(2), we can construct an infinite set of pairs of DL -mates. Then the first question can be answered negatively. It is known (see [5], p. 127) that there are four pairs of A-mates on 10 vertices, one such pair is given in Fig. 3.1 and the complements give another pair, and there are infinite family of cospectral 8-regular graphs (see [13]). This together with Corollary 3.6(2), we can construct an infinite set of pairs of DQ -mates. So the second question also be answered negatively. The Q-cospectral graphs with smallest order are K1,3 and K1,3 (the complement of K1,3 , i.e., K3 ∪ K1 ). We next construct an infinite set of pairs of DQ -cospectral graphs from K1,3 and K1,3 , which also give a negative answer to the second question of Question 3.1. Theorem 3.2. Let G be a connected graph with order k, then G[K1,3 , H2 , . . . , Hk ] and G[K1,3 , H2 , . . . , Hk ] are DQ -mates. k Proof. Let Hi be a graph of order ni (i = 2, . . . , k), and n = 4 + i=2 ni . Set G1 = G[K1,3 , H2 , . . . , Hk ] and G2 = G[K1,3 , H2 , . . . , Hk ]. Then the distance signless Laplacian matrix of G1 is ⎡ ⎤ 2J4×4 + (N1 − 2)I4 − Q(K1,3 ) d12 J4×n2 . . . d1k J4×nk ⎢ ⎥ d12 Jn2 ×4 ⎢ ⎥ DQ (G1 ) = ⎢ ⎥, .. ⎣ ⎦ . ∗ d1k Jnk ×4 the distance signless Laplacian matrix of G2 is ⎡
2J4×4 + (N1 − 2)I4 − Q(K1,3 ) d12 J4×n2 ⎢ d12 Jn2 ×4 ⎢ DQ (G2 ) = ⎢ .. ⎣ . d1k Jnk ×4
. . . d1k J4×nk ∗
⎤ ⎥ ⎥ ⎥. ⎦
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For short, set P = 12 J4×4 − I4 . By straightforward computation, we have P 2 = I4 , P 0 P J4×m = J4×m and P Q(K1,3 )P = Q(K1,3 ). Let R = [ 0 In−4 ], then R2 = In and RDQ (G2 )R = DQ (G1 ). And thus, G1 and G2 are DQ -mates. 2 3.3. Integrality Recall that the D (DL , DQ )-eigenvalues are not only real numbers but also algebraic integers (roots of a monic polynomial with integral coefficients), and for a regular graph, integrality, L-integrality and Q-integrality are equivalent, so from the proof of Corollaries 3.1, 3.2, 3.3, 3.6 and 3.7, we can deduce the following results. Corollary 3.8. If G is a D (DQ )-integral graph of order k 2, H1 , . . . , Hk are r-regular integral graphs with same order, then G[H1 , . . . , Hk ] is D (DQ )-integral graph. Corollary 3.9. If G is a DL -integral graph of order k 2, H1 , . . . , Hk are L-integral graphs with same order, then G[H1 , . . . , Hk ] is DL -integral graph. 3.4. The largest eigenvalue In this subsection, for any connected graph G, we use λD (G) and λDQ (G) to denote the largest D-eigenvalue and DQ -eigenvalue of G, respectively. Proposition 2.2 states that λD (G) and λDQ (G) are always the eigenvalues of the divisor matrix. This together with Corollary 3.5 and Frobenius lower and upper bounds on the eigenvalue of nonnegative matrices (see [14], p. 24), we have the following result. Corollary 3.10. Let G be a connected graph of order k 2, Hi (i = 1, . . . , k) be an ri -regular graph of order ni . Then (1) min1ik {Ni − ri } λD (G[H1 , . . . , Hk ]) max1ik {Ni − ri }, (2) min1ik {2Ni − 2ri } λDQ (G[H1 , . . . , Hk ]) max1ik {2Ni − 2ri }. References [1] K. Balasubramanian, Computer generation of distance polynomials of graphs, J. Comput. Chem. 11 (1990) 829–836. [2] M. Aouchiche, P. Hansen, Two Laplacians for the distance matrix of a graph, Linear Algebra Appl. 439 (2013) 21–33. [3] A.J. Schwenk, Computing the characteristic polynomial of a graph, Graphs Combin. 406 (1974) 153–172. [4] D.M. Cardoso, M.A.A. de Freitas, E.A. Martins, Spectra of graphs obtained by a generalization of the join graph operation, Linear Algebra Appl. 313 (2013) 733–741. [5] D. Cvetković, P. Rowlinson, S. Simić, An Introduction to the Theory of Graph Spectra, first ed., Cambridge University Press, 2010. [6] A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer, Berlin, 2012. [7] H. Fujii, A. Katsuda, Isospectral graphs and isoperimetric constants, Discrete Math. 207 (1999) 33–52.
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[8] W.H. Haemers, E. Spence, Enumeration of cospectral graphs, European J. Combin. 25 (2004) 199–211. [9] L. Halbeisen, N. Hungerbhler, Generation of isospectral graphs, J. Graph Theory 31 (1999) 255–265. [10] R. Merris, Large families of Laplacian isospectral graphs, Linear Multilinear Algebra 43 (1997) 201–205. [11] R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl. 197/198 (1994) 143–176. [12] J. Tan, On isospectral graphs, Interdiscip. Inform. Sci. 4 (1998) 117–124. [13] A. Seress, Large families of cospectral graphs, Des. Codes Cryptogr. 21 (2000) 205–208. [14] H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988.