Spectra of generalized corona of graphs

Linear Algebra and its Applications 493 (2016) 411–425

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Linear Algebra and its Applications www.elsevier.com/locate/laa

Spectra of generalized corona of graphs A.R. Fiuj Laali a , H. Haj Seyyed Javadi a , Dariush Kiani b,c,∗ a

Department of Mathematics and Computer Science, Shahed University, Tehran, Iran b Department of Pure Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), P.O. Box 15875-4413, Tehran, Iran c School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 10 November 2014 Accepted 22 November 2015 Available online xxxx Submitted by R. Brualdi MSC: 05C50 Keywords: Corona Polynomial Cospectral

a b s t r a c t Given simple graphs G, H1 , . . . , Hn , where n = |V (G)|, the n

generalized corona, denoted G˜ ◦ Λ Hi , is the graph obtained i=1 G, H1 ,

by taking one copy of graphs . . . , Hn and joining the ith vertex of G to every vertex of Hi . In this paper, we determine and study the characteristic, Laplacian and signless n

Laplacian polynomial of G˜ ◦ Λ Hi . This leads us to construct i=1

new pairs of cospectral, L-cospectral and Q-cospectral graphs. As an application, we give a simple proof for Csikvari’s Lemma on eigenvalues of graphs. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Throughout the paper In and jn are identity matrix of order n and length-n column ¯ n stands for the graph with n isolated vector consisting entirely of 1’s, respectively. K * Corresponding author at: Department of Pure Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), P.O. Box 15875-4413, Tehran, Iran. E-mail addresses: a.fi[email protected] (A.R. Fiuj Laali), [email protected] (H. Haj Seyyed Javadi), [email protected] (D. Kiani). http://dx.doi.org/10.1016/j.laa.2015.11.032 0024-3795/© 2015 Elsevier Inc. All rights reserved.

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vertices and diag(t1 , . . . , tn ) is a diagonal matrix whose diagonal entries are t1 , . . . , tn . The notation G = ∅ means that the graph G has no vertices and no edges. Let G be a graph with the vertex set {v1 , . . . , vn }. The adjacency matrix of G is an (n × n) matrix A(G) whose (i, j)-entry is 1 if vi is adjacent to vj and 0, otherwise. The characteristic polynomial of G, denoted by fG (λ), is the characteristic polynomial of A(G). We will write it simply fG when there is no confusion. The roots of fG are called the eigenvalues of G. The Laplacian matrix of G and the signless Laplacian matrix of G are defined as L(G) = Δ(G) − A(G) and Q(G) = Δ(G) + A(G), respectively, where Δ(G) is the diagonal matrix whose diagonal entries are degree sequences of G. We denote the Laplacian polynomial of G (L-polynomial) by fL(G) (μ) and the signless Laplacian polynomial of G (Q-polynomial) by fQ(G) (ν). Denote the eigenvalues of A(G), L(G) and Q(G), respectively, by λ1 (G) ≥ λ2 (G) ≥ · · · ≥ λn (G) μ1 (G) ≤ μ2 (G) ≤ · · · ≤ μn (G) ν1 (G) ≤ ν2 (G) ≤ · · · ≤ νn (G). We recall that two graphs are cospectral (L-cospectral and Q-cospectral, respectively) if they have the same characteristic (Laplacian and signless Laplacian, respectively) polynomials. Graph operations are natural techniques for producing new graphs from old ones, and their spectra have received considerable attention in recent years. The corona of G and H, denoted G ◦ H, is the graph obtained by taking one copy of G and |V (G)| copies of H, and joining the ith vertex of G to every vertex in the ith copy of H. This construction was first introduced by Frucht and Harary in [5] with the goal of constructing a graph whose automorphism group is the wreath product of the automorphism group of their components. Since then a number of papers on graph-theoretic properties of corona have appeared. As far as eigenvalues are concerned, the characteristic polynomial, L-polynomial and Q-polynomial of the corona of any two graphs can be expressed by that of two factor graphs [11,10,4,2]. Similarly, the characteristic polynomial, L-polynomial and Q-polynomial of edge corona, neighbourhood corona, subdivision-vertex and subdivision-edge neighbourhood corona of two graphs were completely computed in [6,9,8]. Let G and H be two graphs with n and m vertices, respectively. By a suitable labeling, Cam McLeman and Erin McNicholas in [10] obtained fG◦H (λ) = (fH (λ))n fG (λ − χH (λ)), where χH (λ), coronal of H, is the sum of all entries of the matrix (λI − A(H))−1 T (or χH (λ) = jm (λI − A(H))−1 jm ). Also Qun Liu in [7] obtained fL(G◦H) (μ) = (fL(H) (μ − 1))n fL(G) (μ − m − χL(H) (μ)),

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413

where fL(G) and fL(H) is the Laplacian polynomial of G and H, respectively and χL(H) (μ), L-coronal of H, is the sum of all entries of the matrix ((μ − 1)I − L(H))−1 T (or χL(H) (μ) = jm ((μ − 1)I − L(H))−1 jm ). Similarly, we define Q-coronal of H, the sum of all entries of the matrix ((ν − 1)I − Q(H))−1 , which is denoted by χQ(H) (ν) and we T have that χQ(H) (ν) = jm ((ν − 1)I − Q(H))−1 jm . Let G be a graph with the adjacency matrix A and vertex set V (G) = {1, 2, . . . , n}. Let H1 , . . . , Hn be n graphs (not necessarily non-isomorphic) with adjacency matrices i−1 i−1 i    B1 , . . . , Bn and vertex set V (Hi ) = {n + tk + 1, n + tk + 2, . . . , n + tk } for k=1

k=1

k=1

n

i = 1, 2, . . . , n. By this labeling the adjacency matrix of G˜◦ Λ Hi is given as follows: i=1

 n

A˜◦ Λ Bi := i=1

 A CT

C , D

where C is a matrix of order n × (t1 + · · · + tn ), D is a square matrix of order t1 + · · · + tn and

H1 H2 .. D= . Hn−1 Hn

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

H1

H2

H3

···

Hn

B1 0 .. . 0 0

0 B2 .. . 0 0

0 0 ..

··· ··· .. . Bn−1 0

1 0 2 0 ⎥ ⎥ .. .. ⎥ ⎥, C = . . ⎥ ⎥ n−1 0 ⎦ n Bn

. ··· ···

⎤

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

H1 jtT1 0 .. . 0 0

H2 0 jtT2 .. . 0 0

···

H3 0 0 ..

. ··· 0

··· ··· .. . jtTn−1 0

Hn ⎤ 0 0 ⎥ ⎥ .. ⎥ ⎥ . ⎥ ⎥ 0 ⎦ jtTn

In Section 2 we give some preliminaries. In Section 3 we compute the characteristic k

polynomial of G˜◦ Λ Hi and easily obtain the characteristic polynomial of corona of i=1

two graphs which is published in [10]. The result on the characteristic polynomial of k

G˜ ◦ Λ Hi enables us to construct new pairs of cospectral graphs. As an application, we i=1

give a simple proof for Csikvari’s Lemma [3, Lemma 2.8] on eigenvalues of graphs. In Section 4 and Section 5 we compute the Laplacian and the signless Laplacian polynomial k

of G˜◦ Λ Hi and find the same results as in Section 3. i=1

2. Preliminaries Lemma 2.1 (Schur complement). (See [1].) Let A be an n × n matrix partitioned as  A11 A12 , A21 A22 where A11 and A22 are square matrices. If A11 and A22 are invertible, then

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A.R. Fiuj Laali et al. / Linear Algebra and its Applications 493 (2016) 411–425

 det

A11 A21

A12 A22

= det(A22 ) det(A11 − A12 A22 −1 A21 ) = det(A11 ) det(A22 − A21 A11 −1 A12 )

Notation 2.2. Let G = (P, Q) be a bipartite graph of order n such that |P | = p and |Q| = n − p. Also assume that W is a matrix of order p × (n − p) such that 

0 WT

A(G) =

W 0

.

For  fG (λ) = det

λIp −W T

−W λIn−p

,

with two methods of Schur complement we have: fG (λ) = λn−2p det(λ2 Ip − W W T ) = λn−2p gG (λ) = λ2p−n det(λ2 In−p − W T W ) = λ2p−n qG (λ). It is clear that if n = 2p, then fG (λ) = gG (λ) = qG (λ). The coronal (denoted by χH (λ)) and L-coronal (denoted by χL(H) (μ)) of a graph H are defined [10,7] to be the sum of all entries of the matrices (λI − A(H))−1 and T ((μ − 1)I − L(H))−1 , respectively, or χH (λ) = jm (λI − A(H))−1 jm and χL(H) (μ) = T jm ((μ − 1)I − L(H))−1 jm . Proposition 2.3. (See [10].) Let H be an r-regular graph of order n. Then χH (λ) =

n . λ−r

The proof of the following proposition can be found in the first three lines of the proof of Theorem 3.2 in [7]. Proposition 2.4. (See [7].) Let H be a graph of order n. Then χL(H) (μ) =

n . μ−1

By Proposition 2.4, it is easy to see that if two graphs have the same number of vertices, then they have the same L-coronal. Similarly, we define Q-coronal of the graph H of order n, the sum of all entries of the matrix ((ν − 1)I − Q(H))−1 , which is denoted by χQ(H) (ν) and we have that χQ(H) (ν) = jnT ((ν − 1)I − Q(H))−1 jn .

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415

Proposition 2.5. Let H be an r-regular graph of order n. Then χQ(H) (ν) =

n . ν − 2r − 1

Proof. It is easily seen that ((ν − 1)I − Q(H))jn = ((ν − 1)I − rI − A(H))jn = (ν − 2r − 1)jn 1 ⇒ jn = ((ν − 1)I − rI − A(H))−1 jn ν − 2r − 1 jnT .j = jnT ((ν − 1)I − rI − A(H))−1 jn ν − 2r − 1 n ⇒ χQ(H) (ν) = . 2 ν − 2r − 1 ⇒

Notation 2.6. Let G be a graph of order n. For graphs H1 , . . . , Hn of orders t1 , . . . , tn , respectively, we define the following three n-variable polynomials g(χH1 (λ), . . . , χHn (λ); G) ⎛⎡ λ − χH1 (λ) 0 ⎜⎢ .. = det ⎝⎣ . 0 0 0 Lg(χL(H1 ) (μ), . . . , χL(Hn ) (μ); G) ⎛⎡ μ − t1 − χL(H1 ) (μ) 0 ⎜⎢ .. = det ⎝⎣ . 0 0 0 Qg(χQ(H1 ) (ν), . . . , χQ(Hn ) (ν); G) ⎛⎡ ν − t1 − χQ(H1 ) (ν) 0 ⎜⎢ .. = det ⎝⎣ . 0 0 0

⎤

0

⎞

⎥ ⎟ ⎦ − A(G)⎠ , 0 λ − χHn (λ) ⎤

0

⎞

⎥ ⎟ ⎦ − L(G)⎠ , 0 μ − tn − χL(Hn ) (μ) ⎤

0

⎞

⎥ ⎟ ⎦ − Q(G)⎠ . 0 ν − tn − χQ(Hn ) (ν)

Remark 2.7. Let G be a graph of order n and H1 , . . . , Hn be n graphs. If χH1 (λ) = · · · = χHn (λ) = χH (λ), then g(χH1 (λ), . . . , χHn (λ); G) = fG (λ − χH (λ)). Remark 2.8. Let G be a graph of order n and H1 , . . . , Hn be n graphs, each of them has order m. If χQ(H1 ) (ν) = · · · = χQ(Hn ) (ν) = χQ(H) (ν), then Lg(χL(H1 ) (μ), . . . , χL(Hn ) (μ); G) = fL(G) (μ − m −

m ), μ−1

Qg(χQ(H1 ) (ν), . . . , χQ(Hn ) (ν); G) = fQ(G) (ν − m − χQ(H) (ν)).

A.R. Fiuj Laali et al. / Linear Algebra and its Applications 493 (2016) 411–425

416

k

3. Characteristic polynomial of G˜ ◦ Λ Hi i=1

Theorem 3.1. Let G be a graph of order n and H1 , . . . , Hn be n graphs (not necessarily non-isomorphic) of orders t1 , . . . , tn , respectively. Then f

n

G˜ ◦ Λ Hi

(λ) = (

i=1

n 

fHi (λ)).g(χH1 (λ), . . . , χHn (λ); G).

i=1

Proof. Let A and Bi be the adjacency matrices of G and Hi , respectively, for i = 1, . . . , n. By the labeling given in Section 1, it follows that 

 λI − A C n f (λ) = det , G˜ ◦ Λ Hi λI − D CT i=1 where C is a matrix of order n × (t1 + · · · + tn ) and D is a square matrix of order t1 + · · · + tn and

H1 H2 .. D= . Hn−1 Hn

H1

H2

H3

···

Hn

B1 ⎢ 0 ⎢ ⎢ . ⎢ . ⎢ . ⎢ ⎣ 0 0

0 B2 .. . 0 0

0 0 ..

··· ··· .. . Bn−1 0

1 0 2 0 ⎥ ⎥ .. .. ⎥ ⎥ . . ⎥ ,C= ⎥ ⎦ n−1 0 n Bn

⎡

. ··· ···

⎡

⎤

H1

jtT1 ⎢ 0 ⎢ ⎢ . ⎢ . ⎢ . ⎢ ⎣ 0 0

H2

H3

···

0 jtT2 .. . 0 0

0 0

··· ··· .. . T jtn−1 0

..

. ··· 0

Hn

⎤ 0 0 ⎥ ⎥ .. ⎥ ⎥ . ⎥. ⎥ 0 ⎦ jtTn

By Lemma 2.1, one may obtain that ⎛ f

n

G˜ ◦ Λ Hi

(λ) = (

i=1

n 

i=1

⎡

⎜ ⎢ fHi (λ)). det ⎜ ⎝λI − A − C ⎣ ⎛

=(

n 

⎡

⎜ ⎢ fHi (λ)). det ⎝λI − A − C ⎣

i=1

⎛ n 

⎡

λI − B1

0 ..

0 0 −1

(λI − B1 ) 0 0

0 0 .. .

−1

.

⎞

⎟ ⎥ T ⎦ C ⎟ 0 ⎠ λI − Bn ⎤ ⎞ 0 ⎥ T⎟ ⎦C ⎠ 0 −1

(λI − Bn )

0

jt1 T (λI − B1 )

⎤−1

0

0

0

⎤

⎞

⎥ T⎟ ⎦C ⎠ 0 −1 i=1 0 jtn T (λI − Bn ) ⎤⎞ ⎛ ⎡ T −1 jt1 (λI − B1 ) jt1 0 0 n  ⎥⎟ ⎜ ⎢ .. = ( fHi (λ)). det ⎝λI − A − ⎣ ⎦⎠ . 0 0 −1 i=1 T 0 0 jtn (λI − Bn ) jtn

=(

⎜ ⎢ fHi (λ)). det ⎝λI − A − ⎣

0 0

..

.

A.R. Fiuj Laali et al. / Linear Algebra and its Applications 493 (2016) 411–425

⎛ =(

n 

⎡

χH1 (λ)

⎜ ⎢ fHi (λ)). det ⎝λI − A − ⎣

0 0

i=1

⎛⎡ =(

n 

⎜⎢ fHi (λ)). det ⎝⎣

λ − χH1 (λ) 0 0

i=1

=(

n 

0 .. . 0

0 ..

417

⎤⎞

0

⎥⎟ ⎦⎠ 0 0 χHn (λ) ⎤ ⎞ 0 ⎥ ⎟ ⎦ − A⎠ 0 λ − χHn (λ) .

fHi (λ)).g(χH1 (λ), . . . , χHn (λ); G).

i=1

This completes the proof. 2 The following corollary is the main theorem in [10] which can be easily obtained by Theorem 3.1. Corollary 3.2. Let G and H be two graphs of orders n and m, respectively. Then the following holds: fG◦H (λ) = (fH (λ))n .fG (λ − χH (λ)). Proof. Let H1  . . .  Hn  H. Then by applying Theorem 3.1 and Remark 2.7, we obtain fG◦H (λ) = (fH (λ))n .fG (λ − χH (λ)).

2

Corollary 3.3. Let G be a graph of order n, H1 , . . . , Hn be n graphs and χH1 (λ) = · · · = χHn (λ) = χH (λ). Then f

n

G˜ ◦ Λ Hi

(λ) = (

i=1

n 

fHi (λ)).fG (λ − χH (λ)).

i=1

Proof. It is an immediate consequence of Theorem 3.1 and Remark 2.7.

2

Corollary 3.4. Let G be a graph of order n and H1 , . . . , Hn be r-regular graphs, each of them has order m. Then f

n

G˜ ◦ Λ Hi i=1

(λ) = (

n 

fHi (λ)).fG (λ −

i=1

m ). λ−r

Proof. The proof is straightforward by Proposition 2.3 and Corollary 3.3.

2

Corollary 3.5. Let G1 and G2 be two cospectral graphs of order n and H1 , . . . , Hn be n n

n

i=1

i=1

graphs. If χH1 (λ) = · · · = χHn (λ) = χH (λ), then G1 ˜◦Λ Hi and G2 ˜◦Λ Hi are cospectral.

A.R. Fiuj Laali et al. / Linear Algebra and its Applications 493 (2016) 411–425

418

Proof. It is easily seen that ⎛⎡ f

n

˜ Λ Hi G1 ◦

(λ) = (

i=1

n 

⎜⎢ fHi (λ)). det ⎝⎣

i=1

=(

n 

λ − χH (λ)

0

..

. 0

0 0

fHi (λ)).fG1 (λ − χH (λ)) = (

i=1

=f

0

n 

⎤

⎞

⎥ ⎟ ⎦ − A(G1 )⎠ 0 λ − χH (λ)

fHi (λ)).fG2 (λ − χH (λ))

i=1 n

˜ Λ Hi G2 ◦

2

(λ)

i=1

By Theorem 3.1 the proof of the following corollary is clear. Corollary 3.6. Let G be a graph of order n and H1 , . . . , H2n be a family of cospectral n

2n

i=1

i=n+1

graphs such that χH1 (λ) = · · · = χH2n (λ). Then G˜◦Λ Hi and G˜◦Λ Hi

are cospectral.

Theorem 3.7. Let G = (P, Q) be a bipartite graph of order n, where |P | = p and |Q| = n − p. Let H1  . . .  Hp  Z1 and Hp+1  . . .  Hn  Z2 . Then the following hold: f

n

G˜ ◦ Λ Hi

(λ) =

i=1

⎧  p n−p ⎪ (λ − χZ2 (λ))n−2p gG ( (λ − χZ1 (λ))(λ − χZ2 (λ))) (fZ1 (λ)) .(fZ2 (λ)) ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (f (λ))p .(f (λ))n−p (λ − χ (λ))2p−n q ((λ − χ (λ))(λ − χ (λ))) Z1 Z2 Z1 G Z1 Z2

if

n ≥ 2p

if

n ≤ 2p.

Proof. By the same argument given in the proof of Theorem 3.1 we have ⎛⎡ f

n

G˜ ◦ Λ Hi

(λ) = (

i=1

n 

⎜⎢ fHi (λ)). det ⎝⎣

λ − χH1 (λ)

i=1

 p

n−p

= (fZ1 (λ)) .(fZ2 (λ))

. det

0 0

(λ − χZ1 (λ))Ip 0

0 ..

0

⎤

⎞

⎥ ⎟ ⎦ − A(G)⎠ 0 0 λ − χHn (λ)  0 − A(G) . (λ − χZ2 (λ))In−p .

Since G is bipartite, there is a matrix W of order p × (n − p) such that 

f

n

G˜ ◦ Λ Hi i=1

p

n−p

(λ) = (fZ1 (λ)) .(fZ2 (λ))

(λ − χZ1 (λ))Ip . det −W T

By Lemma 2.1 and Notation 2.2 we obtain that

 −W . (λ − χZ2 (λ))In−p

A.R. Fiuj Laali et al. / Linear Algebra and its Applications 493 (2016) 411–425

f

419

(λ) =

n

G˜ ◦ Λ Hi i=1

⎧  p n−p (fZ1 (λ)) .(fZ2 (λ)) (λ − χZ2 (λ))n−2p gG ( (λ − χZ1 (λ))(λ − χZ2 (λ))) ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

p

n−p

(fZ1 (λ)) .(fZ2 (λ))

(λ − χZ1 (λ))

It is easy to see that χK¯ n (λ) =

2p−n

n λ.

qG (



if

n ≥ 2p

2

(λ − χZ1 (λ))(λ − χZ2 (λ)))

if

n ≤ 2p.

Therefore we have the following corollary.

Corollary 3.8. Let G = (P, Q) be a bipartite graph of order n, where |P | = p and |Q| = ¯ m and Hp+1  . . .  Hn  K ¯ s . Then we have n − p. Let H1  . . .  Hp  K ⎧  ⎪ λmp+(n−p)s (λ − λs )n−2p gG ( (λ − ⎪ ⎪ ⎨ f

n

G˜ ◦ Λ Hi

(λ) =

i=1

⎪ ⎪ ⎪ ⎩ λmp+(n−p)s (λ −



m 2p−n qG ( λ)

(λ −

m λ )(λ

− λs ))

if

n ≥ 2p

m λ )(λ

− λs ))

if n ≤ 2p.

The following corollary is a nice result due to Csikvari [3]. The proof is obtained by Corollary 3.8 for s = 0. Corollary 3.9. Let G = (P, Q) be a bipartite graph of order n, where |P | = p and |Q| = ¯ m and Hp+1 = · · · = Hn = ∅. Then we have n − p. Let H1  . . .  Hp  K ⎧  ⎪ λmp+n−2p gG ( (λ − ⎪ ⎪ ⎨ f

n

G˜ ◦ Λ Hi

(λ) =

i=1

⎪ ⎪ ⎪ ⎩ λmp (λ −

m λ )λ)

 (λ −

m 2p−n qG ( λ)

m λ )λ)

if

n ≥ 2p

if

n ≤ 2p.

k

4. Laplacian polynomial of G˜ ◦ Λ Hi i=1

Notation 4.1. For graphs H1 , . . . , Hn of orders t1 , . . . , tn , respectively. Let V diag(t1 , . . . , tn ). We use this notation in the proof of the following theorem.

=

Theorem 4.2. Let G be a graph of order n and H1 , . . . , Hn be n graphs (not necessarily non-isomorphic) of orders t1 , . . . , tn , respectively. Then

f

n

L(G˜ ◦ Λ Hi ) i=1

(μ) = (

n 

fL(Hi ) (μ − 1)).Lg(χL(H1 ) (μ), . . . , χL(Hn ) (μ); G).

i=1

Proof. Let A and Bi be the adjacency matrices of G and Hi , respectively, for i = 1, . . . , n. By the labeling given in Section 1, it follows that

A.R. Fiuj Laali et al. / Linear Algebra and its Applications 493 (2016) 411–425

420



μI − V − Δ(G) + A C n f (μ) = det T L(G˜ ◦ Λ Hi ) μI − Δ −I +D C i=1 ⎡ ⎢ ⎢ = det ⎢ ⎢ ⎣

⎡ ⎢ ⎢ = det ⎢ ⎢ ⎣

λI − V − Δ(G) + A C

⎢ ⎢ ⎣

T

μI − V − L(G) C

⎡

⎡ ⎢ ⎣

T

C 0 .. .

(μ − 1)I − Δ(H1 ) + B1 0 0

(μ − 1)I − L(H1 ) 0 0

0



0 0 (μ − 1)I − Δ(Hn ) + Bn

⎤ ⎤⎥ ⎥ ⎥⎥ ⎥⎥ ⎦⎦

⎤

C 0 .. . 0

0 0 (μ − 1)I − L(Hn )

⎤⎥ ⎥ , ⎥⎥ ⎦⎥ ⎦

where C and D are the same matrices as we saw in the proof of Theorem 3.1. By Lemma 2.1 we obtain that

f

n

L(G˜ ◦ Λ Hi )

(μ) = (

i=1

n 

fL(Hi ) (μ − 1))

i=1

⎛

⎡

⎜ ⎢ . det ⎜ ⎝μI − V − L(G) − C ⎣

=(

n 

(μ − 1)I − L(H1 )

0 ..

. 0

0 0

⎤−1

0

⎥ ⎦ 0 (μ − 1)I − L(Hn )

⎞ ⎟ CT ⎟ ⎠

fL(Hi ) (μ − 1))

i=1

⎛

⎡

⎜ ⎢ . det ⎝λI − V − L(G) − C ⎣

−1

((μ − 1)I − L(H1 )) 0 0

0

⎛ =(

n 

0 ..

0

.

⎜ fL(Hi ) (μ − 1)). det ⎝ μI − V − L(G)

⎢ −⎣

−1

jt1 T ((μ − 1)I − L(H1 )) 0 0

n 

i=1

0

.

0 ⎛

=(

0 ..

⎤

⎞

⎥ T⎟ ⎦C ⎠ 0 −1 T jtn ((μ − 1)I − L(Hn ))

⎜ fL(Hi ) (μ − 1)). det ⎝ μI − V − L(G)

⎞

⎥ T⎟ ⎦C ⎠ 0 −1 ((μ − 1)I − L(Hn ))

i=1

⎡

⎤

A.R. Fiuj Laali et al. / Linear Algebra and its Applications 493 (2016) 411–425

⎡

−1

jt1 T ((μ − 1)I − L(H1 ))

jt1

0 ..

421

⎤⎞

0

⎥⎟ ⎦⎠ 0 −1 T 0 jtn ((μ − 1)I − L(Hn )) jtn ⎤⎞ ⎛ ⎡ χL(H1 ) (μ) 0 0 n  ⎥⎟ ⎜ ⎢ .. = ( fL(Hi ) (μ − 1)). det ⎝μI − V − L(G) − ⎣ ⎦⎠ . 0 0 i=1 0 0 χL(Hn ) (μ)

⎢ −⎣

0 0

⎛⎡ =(

n 

⎜⎢ fL(Hi ) (μ − 1)). det ⎝⎣

.

μ − t1 − χL(H1 ) (μ) 0 0

i=1

0 ..

⎤

0

.

0

⎞

⎥ ⎟ ⎦ − L(G)⎠ . 0 μ − tn − χL(Hn ) (μ)

By Notation 2.6, it follows that

f

n

L(G˜ ◦ Λ Hi )

(μ) = (

i=1

n 

fL(Hi ) (μ − 1)).Lg(χL(H1 ) (μ), . . . , χL(Hn ) (μ); G).

2

i=1

The following corollary is the main theorem in [7] which can be easily obtained by Theorem 4.2. Corollary 4.3. Let G and H be two graphs of orders n and m, respectively. Then we have  n fL(G◦H) (μ) = fL(H) (μ − 1) .fL(G) (μ − m − χL(H) (μ)). Proof. Let H1  . . .  Hn  H. Then by applying Theorem 4.2 and Remark 2.8, we find that,  n fL(G◦H) (μ) = fL(H) (μ − 1) .fL(G) (μ − m − χL(H) (μ)).

2

Corollary 4.4. Let G be a graph of order n and H1 , . . . , Hn be n graphs each of them has order m. Then f

n

L(G˜ ◦ Λ Hi ) i=1

(μ) = (

n 

fL(Hi ) (μ − 1)).fL(G) (μ − m −

i=1

Proof. It is clear from Theorem 4.2 and Remark 2.8.

m ). μ−1

2

The proofs of the following Corollaries 4.5 and 4.6 are clear from Theorem 4.2. Corollary 4.5. Let G1 and G2 be two L-cospectral graphs of order n and H1 , . . . , Hn be n n n graphs each of them has order m. Then G1 ◦˜ Λ Hi and G2 ˜◦ Λ Hi are L-cospectral. i=1

i=1

A.R. Fiuj Laali et al. / Linear Algebra and its Applications 493 (2016) 411–425

422

Corollary 4.6. Let G be a graph of order n and H1 , . . . , H2n be a family of L-cospectral n

2n

i=1

i=n+1

graphs each of them has order m. Then G˜◦Λ Hi and G˜◦Λ Hi

are L-cospectral.

Theorem 4.7. Let G be an r-regular bipartite graph with |V (G)| = n = 2k, where k is the size of each part. Let H1  . . .  Hk  Z1 and Hk+1  . . .  Hn  Z2 . If |V (Z1 )| = m and |V (Z2 )| = s, then f

n

L(G˜ ◦ Λ Hi )

(μ) = (

i=1

2  



k fL(Zi ) (μ − 1) ).fG (

(μ − m − r −

i=1

s m ).(μ − s − r − )). μ−1 μ−1

Proof. By the same argument given in the proof of Theorem 4.2 we have

f

n

L(G˜ ◦ Λ Hi )

2  k  (μ) = fL(Zi ) (μ − 1)

i=1

i=1

 . det

(μ − m − r − χL(Z1 ) (μ))Ik 0

 0 (μ − s − r − χL(Z2 ) (μ))Ik

+ A(G) .

Since G is bipartite, there is a square matrix W of order k such that f

n

L(G˜ ◦ Λ Hi )

(μ)

i=1

=

2  

k fL(Zi ) (μ − 1) . det

i=1



(μ − m − r − χL(Z1 ) (μ))Ik WT

W (μ − s − r − χL(Z2 ) (μ))Ik

.

By Lemma 2.1, Notation 2.2 and Proposition 2.4 we obtain that f

n

L(G˜ ◦ Λ Hi )

(μ)

i=1

2  

=(

 k fL(Zi ) (μ − 1) ).fG ( (μ − m − r −

i=1

s m ).(μ − s − r − )). μ−1 μ−1

2

The proof of the following corollary is clear from Theorem 4.7. Corollary 4.8. Let G be an r-regular bipartite graph with |V (G)| = n = 2k, where k is ¯ m and Hk+1  . . .  Hn  K ¯ s . Then the the size of each part. Let H1  . . .  Hk  K following holds:  f

n

L(G˜ ◦ Λ Hi ) i=1

(μ) = (μ − 1)k(m+s) .fG (

(μ − m − r −

s m ).(μ − s − r − )). μ−1 μ−1

The proof of the next corollary can be easily obtained by Corollary 4.8 for s = 0.

A.R. Fiuj Laali et al. / Linear Algebra and its Applications 493 (2016) 411–425

423

Corollary 4.9. Let G be an r-regular bipartite graph with |V (G)| = n = 2k, where k is ¯ m and Hk+1 = · · · = Hn = ∅. Then we the size of each part. Let H1  . . .  Hk  K have  m n ).(μ − r)). f (μ) = (μ − 1)km .fG ( (μ − m − r − L(G˜ ◦ Λ Hi ) μ−1 i=1 k

5. Signless Laplacian polynomial of G˜ ◦ Λ Hi i=1

Theorem 5.1. Let G be a graph of order n and H1 , . . . , Hn be n graphs (not necessarily non-isomorphic) of orders t1 , . . . , tn , respectively. Then f

n

Q(G˜ ◦ Λ Hi )

(ν) = (

i=1

n 

fQ(Hi ) (ν − 1)).Qg(χQ(H1 ) (ν), . . . , χQ(Hn ) (ν); G).

i=1

Proof. The proof is similar to Theorem 4.2.

2

Corollary 5.2. Let G and H be two graphs of orders n and m, respectively. Then we have  n fQ(G◦H) (ν) = fQ(H) (ν − 1) .fQ(G) (ν − m − χQ(H) (ν)). Proof. Let H1  . . .  Hn  H. Then by applying Theorem 5.1 and Remark 2.8, we obtain that  n fQ(G◦H) (ν) = fQ(H) (ν − 1) .fQ(G) (ν − m − χQ(H) (ν)).

2

Corollary 5.3. Let G be a graph of order n and H1 , . . . , Hn be n graphs each of them has order m such that χQ(H1 ) (ν) = · · · = χQ(Hn ) (ν) = χQ(H) (ν). Then f

n

Q(G˜ ◦ Λ Hi )

(ν) = (

i=1

n 

fQ(Hi ) (ν − 1)).fQ(G) (ν − m − χQ(H) (ν))

i=1

Proof. It is clear from Theorem 5.1 and Remark 2.8.

2

Corollary 5.4. Let G be a graph of order n and H1 , . . . , Hn be r-regular graphs each of them has order m. Then f

n

Q(G˜ ◦ Λ Hi ) i=1

(ν) = (

n 

i=1

fQ(Hi ) (ν − 1)).fQ(G) (ν − m −

m ). ν − 2r − 1

Proof. The proof is straightforward by Proposition 2.5 and Corollary 5.3. The proofs of two following corollaries are clear from Theorem 5.1.

2

A.R. Fiuj Laali et al. / Linear Algebra and its Applications 493 (2016) 411–425

424

Corollary 5.5. Let G1 and G2 be two Q-cospectral graphs of order n and H1 , . . . , Hn be n n graphs each of them has order m and χQ(H1 ) (ν) = · · · = χQ(Hn ) (ν). Then G1 ◦˜ Λ Hi i=1

n

and G2 ˜◦ Λ Hi are Q-cospectral. i=1

Corollary 5.6. Let G be a graph of order n and H1 , . . . , H2n be a family of Q-cospectral n

graphs each of them has order m such that χQ(H1 ) (ν) = · · · = χQ(Hn ) (ν). Then G˜◦Λ Hi

i=1

2n G˜◦Λ Hi i=n+1

and

are Q-cospectral.

Theorem 5.7. Let G be an r-regular bipartite graph with |V (G)| = n = 2k, where k is the size of each part. Let H1  . . .  Hk  Z1 and Hk+1  . . .  Hn  Z2 . If |V (Z1 )| = m and |V (Z2 )| = s, then f

n

Q(G˜ ◦ Λ Hi )

(ν)

i=1

=(

2  

 k fQ(Zi ) (ν − 1) ).fG ( (ν − m − r − χQ(Z1 ) (ν)).(ν − s − r − χQ(Z2 ) (ν))).

i=1

Proof. The proof is the same as the proof of Theorem 4.7. It is easy to see that χQ(K¯ n ) (ν) =

n ν−1 .

2

Therefore we obtain the following corollary.

Corollary 5.8. Let G be an r-regular bipartite graph with |V (G)| = n = 2k, where k is ¯ m and Hk+1  . . .  Hn  K ¯ s . Then the the size of each part. Let H1  . . .  Hk  K following holds:  f

n

Q(G˜ ◦ Λ Hi )

(ν) = (ν − 1)k(m+s) .fG (

(ν − m − r −

i=1

s m ).(ν − s − r − )). ν−1 ν−1

The proof of the following corollary can be easily obtained by Corollary 5.8 for s = 0. Corollary 5.9. Let G be an r-regular bipartite graph with |V (G)| = n = 2k, where k is ¯ m and Hk+1 = · · · = Hn = ∅. Then the the size of each part. Let H1  . . .  Hk  K following holds:  f

n

Q(G˜ ◦ Λ Hi ) i=1

(ν) = (ν − 1)

km

.fG (

(ν − m − r −

m ).(ν − r)). ν−1

Acknowledgments The authors would like to thank the referees for their helpful comments and suggestions. The research of the third author was in part supported by a grant from the Institute for Research in Fundamental Sciences (IPM) (Grant No. 94050116).

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