SPECTRA OF GRAPH OPERATIONS BASED ON CORONA AND NEIGHBORHOOD CORONA OF GRAPH G AND K 1

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1 JOURNAL OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) , ISSN (o) / JOURNALS / JOURNAL Vol. 5(2015), Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN (p) , ISSN (o) X SPECTRA OF GRAPH OPERATIONS BASED ON CORONA AND NEIGHBORHOOD CORONA OF GRAPH G AND K 1 Chadrashekar Adiga ad Rakshith B. R. Abstract. The coroa G H of two graphs G ad H is the graph obtaied by takig oe copy of G ad V (G) copies of H ad joiig each i-th vertex of G to every vertex i the i-th copy of H. The eighborhood coroa G H of two graphs G ad H is the graph obtaied by takig oe copy of G ad V (G) copies of H ad joiig the eighbors of the i-th vertex of G to every vertex i the i-th copy of H. I this paper we compute the adjacecy spectrum (respectively, Laplacia spectrum, sigless Laplacia) of four types of graph operatios o G ad H, called as N- vertex coroa, C-vertex eighborhood coroa, C-edge coroa ad N-edge coroa, based o the coroa ad eighborhood coroa of G ad K 1. As a applicatio, our results eable us to costruct ifiitely may pairs of cospectral graphs ad also itegral graphs. 1. Itroductio All graphs cosidered i this paper are simple graphs. Let G be a graph with vertex set V (G) = {v 1, v 2,, v } ad edge set E(G). The adjacecy matrix of G, deoted by A(G), is the matrix [a ij ], where a ij = 1, if the vertices v i ad v j are adjacet i G, otherwise a ij = 0. If all the eigevalues of A(G) are itegers the the graph G is said to be a itegral graph [10]. The Laplacia matrix of the graph G, deoted by L(G), is defied as D(G) - A(G), where D(G) is the diagoal degree matrix of G. The sigless Laplacia matrix of the graph G, deoted by Q(G) is defied as D(G)+A(G). The adjacecy spectrum σ(g), Laplacia spectrum µ(g), 2010 Mathematics Subject Classificatio. 05C50. Key words ad phrases. Spectrum, coroa, eighbourhood coroa, cospectral graphs, itegral graphs. 55

2 56 CHANDRASHEKAR ADIGA AND RAKSHITH B. R. ad sigless Laplacia spectrum γ(g) are defied as follows: σ(g) = (λ 1 (G), λ 2 (G),, λ (G)), µ(g) = (µ 1 (G), µ 2 (G),, µ (G)), γ(g) = (γ 1 (G), γ 2 (G),, γ (G)), where λ i (G), µ i (G) ad γ i (G) are the eigevalues of A(G), L(G) ad Q(G), respectively. Also ad λ 1 (G) λ 2 (G) λ (G), µ 1 (G) = 0 µ 2 (G) µ (G), γ 1 (G) γ 2 (G) γ (G). Studies o differet spectra of graphs ca be foud i [1, 5, 6, 7, 8, 15, 19] ad therei refereces. Two graphs are said to be adjacecy cospectral (Laplacia cospectral, sigless Laplacia cospectral, respectively) if they have the same adjacecy spectrum (Laplacia spectrum, sigless Laplacia spectrum, respectively). The coroa G H of two graphs G ad H is the graph obtaied by takig oe copy of G ad V (G) copies of H ad joiig each i-th vertex of G to every vertex i the i-th copy of H. The coroa of two graphs was first itroduced by Frucht ad Harary i [9]. The eighborhood coroa G H of two graphs G ad H is the graph obtaied by takig oe copy of G ad V (G) copies of H ad joiig the eighbors of the i-th vertex of G to every vertex i the i-th copy of H. The eighborhood coroa of two graphs was itroduced by Idulal [12]. More iformatio about the coroa ad eighborhood coroa ca be foud i [3, 4, 12, 17]. Several graph operatios such as disjoit uio, NEPS, coroa, edge coroa, eighborhood coroa, subdivisio vertex coroa, subdivisio edge coroa, subdivisio eighborhood coroa, etc., have bee itroduced ad their spectrum were studied by various Mathematicias. Details about the spectra of some ew graph operatios ca be foud i [3, 11, 12, 16, 17, 18]. Recetly, La ad Zhou [14] have itroduced four ew graph operatios called as R-vertex coroa, R-edge coroa, R-vertex eighborhood coroa ad R-edge eighborhood coroa ad provided a complete iformatio about the spectra of these four graph operatios. Motivated by the above works, we defie four ew graph operatios called as N- vertex coroa, C-vertex eighborhood coroa, C-edge coroa ad N-edge coroa, based o the coroa ad eighborhood coroa of a graph G ad K 1. Further we compute their spectrum i some cases. The paper is orgaized as follows: I Sectio 3, we give the adjacecy spectra (respectively, Laplacia spectra, sigless Laplacia spectra) of N- vertex coroa ad C-vertex eighborhood coroa of two graphs G ad H. I Sectio 4, we give the adjacecy spectra (respectively, Laplacia spectra, sigless Laplacia spectra) of C-edge coroa ad N-edge coroa of two graphs G ad H. I Sectio 5, usig the results obtaied i Sectios 3 ad 5 we give methods to costruct ifiitely may pairs of cospectral graphs ad also itegral graphs.

3 SPECTRA OF GRAPH OPERATIONS BASED ON CORONA Prelimiaries I this sectio, we give some defiitios ad lemmas which are useful to prove our mai results. Let G 1 ad G 2 be two graphs o ad m vertices respectively. Let N(G 1 ) = G 1 K 1, C(G 1 ) = G 1 K 1. We defie four ew graph operatios o G 1 ad G 2 as follows: Defiitio 2.1. The N-vertex coroa G 1 G 2 of two graphs G 1 ad G 2 is the graph obtaied by takig oe copy of N(G 1 ), V (G 1 ) copies of G 2 ad joiig each i-th vertex of G 1 to every vertex of the i-th copy of G 2. Defiitio 2.2. The C-vertex eighborhood coroa G 1 G 2 of two graphs G 1 ad G 2 is the graph obtaied by takig oe copy of C(G 1 ), V (G 1 ) copies of G 2 ad joiig each eighbors of the i-th vertex of G 1 to every vertex of i-th copy of G 2. Defiitio 2.3. The N-edge coroa G 1 G 2 of two graphs G 1 ad G 2 is the graph obtaied by takig oe copy of N(G 1 ), E(G 1 ) copies of G 2 ad joiig each termial vertex of i-th edge of G 1 to every vertex of the i-th copy of G 2. Defiitio 2.4. The C-edge coroa G 1 G 2 of two graphs G 1 ad G 2 is the graph obtaied by takig oe copy of C(G 1 ), E(G 1 ) copies of G 2 ad joiig each termial vertex of i-th edge of G 1 to every vertex of i-th copy of G 2. Let A = (a ij ) be a m matrix, B = (b ij ) be a p q matrix the the Kroecker product [7] A B of A ad B is the p by mq matrix obtaied by replacig each etry a ij of A by a ij B. It is well-kow that (A B)(C D) = AC BD, wheever the products AC ad BD exist. The M-coroal Γ M (x) of a square matrix M of order [4, 18] is defied as follows: Γ M (x) = e T (xi M) 1 e, where e is the colum vector of size with all its etries are 1. If M is a square matrix of order such that sum of etries i each row a costat r the it is easy to see that Γ M (x) = /(x r). Further for a complete bipartite graph K p,q we have [4] Γ A(Kp,q )(x) = (p + q)x + 2pq x 2. pq Lemma 2.1. [7] If M, N, P, Q are matrices with M beig a o-sigular matrix, the M N P Q = M Q P M 1 N.

4 58 CHANDRASHEKAR ADIGA AND RAKSHITH B. R. 3. Spectra of N-vertex coroa ad C-vertex eighborhood coroa I this sectio, we determie the characteristic polyomial of N- vertex coroa ad C-vertex eighborhood coroa of two graphs G 1 ad G 2 i terms of coroal of a matrix. Also we compute the adjacecy spectrum (respectively, Laplacia spectrum, sigless Laplacia spectrum) of C- edge coroa ad N-edge coroa of two graphs G 1 ad G 2 i some cases. Theorem 3.1. Let G 1 ad G 2 be two graphs o ad m vertices respectively. The f(a(g 1 G 2 ), x) = m (x λ i (G 2 )) (x Γ (x) λ A(G2 ) i(g 1 ))x λ 2 i (G 1 ). Proof. With suitable labellig of the vertices of G, the adjacecy matrix A(G 1 G 2) ca be formulated as follows: I A(G 2 ) 0 I e A(G 1 G 2 ) = 0 0 A(G 1 ), I e T A(G 1 ) A(G 1 ) where e is the colum vector of size m, with all its etries are 1, I is the idetity matrix of order. Now, I (xi m A(G 2 )) 0 I e f(a(g 1 G 2 ), x) = det 0 xi A(G 1 ) I e T A(G 1 ) xi A(G 1 ) By Lemma 2.1, we have (3.1) f(a(g 1 G 2 ), x) = where m (x λ i (G 2 )) dets, xi A(G 1 ) S = A(G 1 ) (x Γ (x))i A(G2 ) A(G 1 ) Agai usig Lemma 2.1, we see that dets = x det((x Γ (x))i A(G2 ) A(G 1 ) A 2 (G 1 )/x) (3.2) = (x Γ (x) λ A(G2 ) i(g 1 ))x λ 2 i (G 1 ). So by (3.1) ad (3.2) the result follows.

5 SPECTRA OF GRAPH OPERATIONS BASED ON CORONA 59 As Γ M (x) =, where M is the square matrix of order with each of its (x r) (p + q)x + 2pq row sum a costat r ad Γ Kp,q (x) = x 2, proofs of the followig two pq corollaries follows immediately by the above theorem. Corollary 3.1. Let G 1 be a arbitrary graph ad G 2 be a r-regular graph o ad m vertices respectively. The the adjacecy spectrum of G = (G 1 G 2 ) is give by: a. λ i (G 2 ), with multiplicity for i = 2,, m. b. three roots of the polyomial x 3 (r + λ i (G 1 ))x 2 + (rλ i (G 1 ) m λ 2 i (G 1))x + rλ 2 i (G 1), for i = 1,,. Corollary 3.2. Let G 1 be a arbitrary graph o vertices. The the adjacecy spectrum of G 1 K p,q is give by: (a) 0 with multiplicity (p + q 2). (b) four roots of the polyomial x 4 λ i(g 1)x 3 ( λ 2 i (G 1) + pq + p + q ) x 2 + (λ i(g 1) 2) pqx + λ 2 i (G 1)pq, for i = 1,,. Theorem 3.2. Let G 1 be r 1 -regular o vertices ad G 2 be a arbitrary graph o m vertices. The the Laplacia spectrum of G 1 G 2 is give by: a. µ i (G 2 ) + 1 with multiplicity, for i = 2,, m. b. three roots of the polyomial x 3 (µ i(g 1) + m + 2 r 1 + 1) x 2 ( µ i(g 1) 2 3 µ i(g 1)r 1 mr 1 µ i(g 1) 2 r 1 ) x + µ i(g 1) 2 3 µ i(g 1)r 1, for i = 1,,. Proof. With suitable labellig of the vertices of G, the adjacecy matrix L(G 1 G 2 ) ca be formulated as follows: I (I m + L(G 2 )) 0 I e L(G 1 G 2 ) = 0 r 1 I A(G 1 ), I e T A(G 1 ) (r 1 + m)i + L(G 1 ) where e is the colum vector of size m with all its etries are 1, I is the idetity matrix of order. Now, I ((x 1)I m L(G 2)) 0 I e f(l(g 1 G 2 )) = det 0 (x r 1)I A(G 1) By Lemma 2.1, it follows that (3.3) f(l(g 1 G 2 )) = I e T A(G 1) (x r 1 m)i L(G 1) m (x µ i (G 2 ) 1) dets,

6 60 CHANDRASHEKAR ADIGA AND RAKSHITH B. R. where (x r 1 )I A(G 1 ) S = A(G 1 ) (x Γ (x 1) r L(G2 ) 1 m)i L(G 1 ) Agai usig Lemma 2.1, we see that dets = (x r 1 ) det((x Γ (x 1) r L(G2 ) 1 m)i L(G 1 ) A 2 (G 1 )/(x r 1 )) (3.4) = (x m/(x 1) r 1 m µ i (G 1 ))(x r 1 ) (µ i (G 1 ) r 1 ) 2. So, by (3.3) ad (3.4) the desired result follows. Let t(g) deote the umber of spaig trees of G. It is well kow [7] that for a coected graph G o vertices, t(g) is give by (3.5) t(g) = µ 2(G) µ (G). Corollary 3.3. Let G 1 be r 1 -regular graph o vertices ad G 2 be a arbitrary graph o m vertices. The the umber of spaig trees of G 1 G 2 is give by m t(g 1 G 2 ) = r 1 t(g 1 ) (3r 1 µ i (G 1 )) (µ i (G 2 ) + 1). i=2 i=2 Proof. Proof follows directly from the above theorem ad (3.5). Theorem 3.3. Let G 1 be r 1 -regular o vertices ad G 2 be r 2 -regular graph o m vertices. The the sigless Laplacia spectrum of G 1 G 2 is give by: a. γ i (G 2 ) + 1 with multiplicity, for i = 2,, m. b. three roots of the polyomial (x 2r 2 1)(x r 1 )(x r 1 m γ i (G 1 )) m(x r 1 ) (γ i (G 1 ) r 1 ) 2 (x 2r 2 1), for i = 1,,. Proof. With suitable labellig of the vertices of G, the adjacecy matrix Q(G 1 G 2 ) ca be formulated as follows: I (I m + Q(G 2 )) 0 I e Q(G 1 G 2 ) = 0 r 1 I A(G 1 ), I e T A(G 1 ) (r 1 + m)i + Q(G 1 ) where e is the colum vector of size m with all its etries are 1, I is the idetity matrix of order. Rest of the proof is similar to the proof of Theorem 3.2. As the proofs of the followig theorems are similar to that of Theorem 3.1, we omit the details.

7 SPECTRA OF GRAPH OPERATIONS BASED ON CORONA 61 Theorem 3.4. Let G 1 ad G 2 be two graphs o ad m vertices. The f(a(g 1 G 2 ), x) = m (x λ i (G 2 )) (x 2 (λ i (G 1 ) + λ 2 i (G 1 )Γ A(G2 (x))x 1). ) Usig the fact that Γ M (x) =, where M is the square matrix of order (x r) (p + q)x + 2pq with each of its row sum a costat r ad Γ Kp,q (x) = x 2, i the pq above theorem we have the followig two corollaries. Corollary 3.4. Let G 1 be a arbitrary graph ad G 2 be a r-regular graph, o ad m vertices, respectively. The the adjacecy spectrum of G 1 G 2 is give by: (a) λ i (G 2 ) with multiplicity, for i = 2,, m. (b) three roots of the polyomial x 3 (λ i (G 1 ) + r)x 2 (λ 2 i (G 1)m λ i (G 1 )r + 1))x + r, for i = 1,,. Corollary 3.5. Let G 1 be a arbitrary graph o vertices. The the adjacecy spectrum of G 1 K p,q is give by: (a) 0 with multiplicity (p + q 2). (b) four roots of the polyomial x 4 λ i (G 1 )x 3 ( λ i (G 1 ) 2 p + λ i (G 1 ) 2 q + pq + 1 ) x 2 + ( 2 λ i (G 1 ) 2 + λ i (G 1 ) ) pqx+pq, for i = 1,,. Theorem 3.5. Let G 1 be r 1 -regular ad G 2 be a arbitrary graph o ad m vertices respectively. The the Laplacia spectrum of G 1 G 2 is give by: a. µ i (G 2 ) + r 1 with multiplicity, for i = 2,, m. b. three roots of the polyomial x 3 (mr 1 + λ i(g 1) + r 1 + 2) x 2 +(( λ i(g 1)m + 2 mr 1 + r 1 + 1)λ i(g 1) + mr r 1) x + (λ i (G 1 )m 2 mr 1 r 1 )λ i (G 1 ), for i = 1,,. By the above theorem ad (3.5), we have the followig corollary: Corollary 3.6. Let G 1 be a r 1 -regular graph ad G 2 be a arbitrary graph o ad m vertices, respectively. The the umber of spaig trees of G 1 G 2 is give by: m t(g 1 G 2 ) = r 1 t(g 1 ) (2mr 1 µ i (G 1 ) + r 1 ) (µ i (G 2 ) + r 1 ). i=2 Theorem 3.6. Let G 1 be r 1 -regular ad G 2 be r 2 -regular graph o ad m vertices, respectively. The the sigless Laplacia spectrum of G 1 G 2 is give by: a. γ i (G 2 ) + r 1 with multiplicity, for i = 2,, m. b. three roots of the polyomial x 3 (mr 1 + γ i (G 1 ) + 2 r 2 + r 1 + 2) x 2 + (( γ i (G 1 )m + 2 mr r 2 + r 1 + 1)γ i (G 1 ) +mr r r r 1mr 2) x + γ i(g 1) 2 m 2 γ i(g 1)mr 1 2 r 1mr 2 2 γ i(g 1)r 2 γ i (G 1 )r 1, for i = 1,,. i=2

8 62 CHANDRASHEKAR ADIGA AND RAKSHITH B. R. 4. Spectra of C-edge coroa ad N-edge coroa I this sectio, we determie the characteristic polyomial of C- edge coroa ad N-edge coroa of two graphs G 1 ad G 2 i terms of coroal of a matrix. Also we compute the adjacecy spectrum (respectively, Laplacia spectrum, sigless Laplacia spectrum) of C- edge coroa ad N-edge coroa of two graphs G 1 ad G 2 i some cases. Theorem 4.1. Let G i (i = 1, 2) be r i - regular graphs with i vertices ad m i edges, respectively. The 2 1 f(a(g 1 G 2), x) = (x λ i(g 2)) m 1 (x 2 (λ i(g 1) + (λ i(g 1) + r 1)Γ A(G2 (x))x 1). ) Proof. With suitable labellig of the vertices of G 1 G 2, the adjacecy matrix A(G 1 G 2 ) ca be formulated as follows: I m1 A(G 2 ) 0 B e A(G 1 G 2 ) = 0 0 I 1, B T e T I 1 A(G 1 ) where e is the colum vector of size 2 with all its etries are 1, I 1 is the idetity matrix of order 1 ad B is the icidece matrix of G 1. Now, I m1 (xi 2 A(G 2 )) 0 B e f(a(g 1 G 2 )) = det 0 xi 1 I 1 B T e T I 1 xi 1 A(G 1 ) By Lemma 2.1, we have 2 (4.1) f(a(g 1 G 2 )) = (x λ i (G 2 )) m1 dets, where S = xi 1 I 1 I 1 xi 1 A(G 1 ) (A(G 1 ) + r 1 I 1 )Γ A(G2 ) (x) Agai employig Lemma 2.1 to dets, we see that dets = x 1 det(xi 1 A(G 1 ) (A(G 1 ) + r 1 I 1 )Γ (x) I A(G2 ) 1 /x) 1 (4.2) = (x 2 (λ i (G 1 ) + (λ i (G 1 ) + r 1 )Γ A(G2 (x))x 1). ) So by (4.1) ad (4.2) the result follows.

9 SPECTRA OF GRAPH OPERATIONS BASED ON CORONA 63 Usig the fact that Γ M (x) =, where M is the square matrix of order (x r) (p + q)x + 2pq with each of its row sum a costat r ad Γ Kp,q (x) = x 2, i the pq above theorem we have the followig two corollaries. Corollary 4.1. Let G i (i = 1, 2) be r i -regular graphs with i vertices ad m i edges, respectively. The the adjacecy spectrum of G 1 G 2 is give by: (a) λ i (G 2 ) with multiplicity m 1, for i = 2,, 2. (b) r 2 with multiplicity m 1 1. (c) three roots of the polyomial x 3 (λ i (G 1 ) + r 2 ) x 2 + (λ i (G 1 )r 2 2 λ i (G 1 ) r 1 2 1) x + r 2, for i = 1,, 1. Corollary 4.2. Let G 1 be r 1 -regular graph (r 1 2) with 1 vertices ad m 1 edges. The the adjacecy spectrum of G 1 K p,q is give by: (a) 0 with multiplicity m 1 (p + q 2). (b) ± pq with multiplicity m 1 1. (c) four roots of the polyomial x 4 λ i(g 1)x 3 (λ i(g 1)p + λ i(g 1)q + pq + pr 1 + qr 1 + 1) x 2 pq (λ i(g 1) + 2 r 1) x+pq, for i = 1,, 1. Theorem 4.2. Let G 1 be r 1 - regular graph (r 1 2) ad G 2 be a arbitrary graph with 1, 2 vertices ad m 1, m 2 edges, respectively. The the Laplacia spectrum of G 1 G 2 is give by: a. µ i (G 2 ) + 2 with multiplicity m 1, for i = 2,, 2. b. 2 with multiplicity m 1 1. c. three roots of the polyomial (x 1)((x 2)(x 2 r 1 µ i (G 1 ) 1) + (µ i (G 1 ) 2r 1 ) 2 ) (x 2), for i = 1,, 1. Proof. With suitable labellig of the vertices of G 1 G 2, the adjacecy matrix L(G 1 G 2 ) ca be formulated as follows: I m1 (2I 2 + L(G 2 )) 0 B e L(G 1 G 2 ) = 0 I 1 I 1, B T e T I 1 (mr 1 + 1)I 1 + L(G 1 ) where e is the colum vector of size 2 with all its etries are 1, I 1 is the idetity matrix of order 1 ad B is the icidece matrix of G 1. Now, I m1 ((x 2)I 2 L(G 2)) 0 B e f(l(g 1 G 2 )) = det 0 (x 1)I 1 I 1 By Lemma 2.1, we have B T e T I 1 (x mr 1 1)I 1 L(G 1)

10 64 CHANDRASHEKAR ADIGA AND RAKSHITH B. R. 2 (4.3) f(l(g 1 G 2 )) = (x µ i (G 2 ) 2) m1 dets, where (x 1)I 1 I 1 S = I 1 (x mr 1 1)I 1 L(G 1 ) (A(G 1 ) + r 1 I 1 )Γ L(G2 (x 2) ) Agai employig Lemma 2.1 to dets, we see that dets = (x 1) 1 det((x 2r 1 1)I 1 L(G 1) (A(G 1) + r 1I 1 )Γ (x 2) L(G2 ) I 1 /(x 1)) 1 (4.4) = (x 1)(x 2r 1 µ i(g 1) 1 + (µ i(g 1) 2r 1) 2/(x 2)) 1. So by (4.3) ad (4.4) the result follows. Corollary 4.3. Let G 1 be r 1 - regular graph (r 1 2) ad G 2 be a arbitrary graph with 1, 2 vertices ad m 1, m 2 edges, respectively. The the umber of spaig trees of G 1 G 2 is give by: 2 t(g 1 G 2 ) = t(g 1 )2 m ( 2 + 2) 1 (µ i (G 2 ) + 2) m 1. Proof. By (3.5) ad above theorem, we obtai the desired result. i=2 Theorem 4.3. Let G 1 be r 1 - regular graph (r 1 2) ad G 2 be a r 1 graph with 1, 2 vertices ad m 1, m 2 edges. The the sigless Laplacia spectrum of G 1 G 2 is give by: a. γ i (G 2 ) + 2 with multiplicity m 1, for i = 2,, 2. b. 2(r 2 + 1) with multiplicity m 1 1. c. three roots of the polyomial x 3 (r γ i (G 1 ) + 2 r 2 + 4) x 2 + (2 r 1 2 r γ i (G 1 )r 2 γ i (G 1 ) r γ i (G 1 ) + 4 r 2 + 4) x 2 r 1 2 r 2 2 γ i (G 1 )r 2 + γ i (G 1 ) 2 2 r γ i (G 1 ), for i = 1,, 1. Proof. With suitable labellig of the vertices of G 1 G 2, the adjacecy matrix Q(G 1 G 2 ) ca be formulated as follows: I m1 (2I 2 + Q(G 2 )) 0 B e Q(G 1 G 2 ) = 0 I 1 I 1, B T e T I 1 (mr 1 + 1)I 1 + Q(G 1 ) where e is the colum vector of size 2 with all its etries are 1, I 1 is the idetity matrix of order 1 ad B is the icidece matrix of G 1. Rest of the proof is similar to the proof of Theorem 4.2.

11 SPECTRA OF GRAPH OPERATIONS BASED ON CORONA 65 As the proofs of the followig theorems are similar to that of Theorem 4.1 we omit the details. Theorem 4.4. Let G i (,2) be r i - regular graphs with i vertices ad m i edges, respectively. The 2 f(a(g 1 G 2 ), x) = (x λ i (G 2 )) m 1 (x (λ i (G 1 )+r 1 )Γ (x) λ A(G2 ) i(g 1 ))x λ 2 i (G 1 ). As Γ M (x) =, where M is the square matrix of order with each of its (x r) (p + q)x + 2pq row sum a costat r ad Γ Kp,q (x) = x 2, proofs of the followig two pq corollaries follows immediately by the above theorem. Corollary 4.4. Let G i (i = 1, 2) be r i -regular graphs with r 1 2, i vertices ad m i edges. The the adjacecy spectrum of G = (G 1 G 2 ) is give by: a. λ i (G 2 ), with multiplicity m 1 for i = 2,, 2. b. r 2 with multiplicity m 1 1. c. three roots of the polyomial x 3 (r 2 + λ i (G 1 ))x 2 (λ 2 i (G 1) + λ i (G 1 )( 2 r 2 ) + 2 r 1 )x + r 2 λ 2 i (G 1), for i = 1,, 1. Corollary 4.5. Let G 1 be r 1 -regular graphs with 1 vertices ad m 1 edges. The the adjacecy spectrum of G = (G 1 K p,q ) is give by: a. 0, with multiplicity m 1 (p + q 2). b. ± pq with multiplicity m 1 1. c. four roots of the polyomial ) x 4 λ i (G 1 )x 3 + ( λ i (G 1 ) 2 + ( p q) λ i (G 1 ) + ( q r 1 ) p qr 1 x 2 pq (λ i (G 1 ) + 2 r 1 ) x + λ i (G 1 ) 2 pq, for i = 1,, 1. Theorem 4.5. Let G 1 be r 1 -regular graph (r 1 2) with 1 vertices ad m 1 edges. The for a arbitrary graph G 2 o 2 vertices, the Laplacia spectrum of G 1 G 2 is give by: a. µ i (G 2 ) + 2 with multiplicity m 1, for i = 2,, 2. b. 2 with multiplicity m 1 1. c. three roots of the polyomial (x 2)(x r 1 )(x µ i (G 1 ) r 1 2 r 1 ) + 2 (µ i (G 1 ) 2r 1 )(x r 1 ) (µ i (G 1 ) r 1 ) 2 (x 2), for i = 1,, 1. Applyig (3.5) to the above theorem we have the followig result: Corollary 4.6. Let G 1 be r 1 -regular graph (r 1 2) with 1 vertices ad m 1 edges. The for a arbitrary graph G 2 o 2 vertices, the umber of spaig trees of G 1 G 2 is give by: 1 2 t(g 1 G 2 ) = r 1 t(g 1 )2 m (6r r 1 2µ i (G 1 )) (µ i (G 2 ) + 2) m 1. i=2 i=2

12 66 CHANDRASHEKAR ADIGA AND RAKSHITH B. R. Theorem 4.6. Let G i (i = 1, 2) be r i -regular graphs with i vertices, m i edges ad r 1 2. The the sigless Laplacia spectrum of G 1 G 2 is give by: a. γ i (G 2 ) + 2 with multiplicity m 1, for i = 2,, 2. b. 2(r 2 + 1) with multiplicity m 1 1. c. three roots of the polyomial (x 2r 2 2)(x r 1 )(x γ i (G 1 ) r 1 2 r 1 ) 2 γ i (G 1 )(x r 1 ) (γ i (G 1 ) r 1 ) 2 (x 2r 2 2), for i = 1,, Applicatios The otio of itegral graph was first itroduced by Harary ad Schwek i 1974 [10]. I geeral, the problem of characterizig itegral graphs is a difficult task. I [2] costructios ad properties of itegral graphs are discussed i detail. The graphs K, K m, (m a perfect square), C 6, the cocktail parity graph CP () = K 2, are all examples of itegral graphs. Moreover, some graph operatios such as cartesia product, direct product ad strog product whe applied o itegral graphs produce agai itegral graphs. For other related works see [13, 20, 21] ad therei refereces. I this sectio, we apply our graph operatios based o coroa ad eighborhood coroa o kow itegral graphs to produce class of ew itegral graphs. At the ed of the sectio, as a applicatio we give methods to costruct ifiitely may pairs of cospectral graphs. From Corollaries 3.1 ad 3.4 it follows that a. If G is a itegral graph of order, the G mk 1 is itegral if ad oly if 5λ 2 i (G) + 4m is a perfect square, for i = 1, 2,,. b. If G is a itegral graph of order, the G mk 1 is itegral if ad oly if λ 2 i (G)(4m + 1) + 4 is a perfect square, for all i = 1, 2,,. I particular, we have the followig: i. K mk 1 is itegral if ad oly if 4m+5 ad m + 5 are perfect squares. ii. K p,q mk 1 is itegral if ad oly if pq, m ad 5pq + 4m are perfect squares. iii. If G mk 1 is a itegral graph the (K 2 G) mk 1 is itegral, where deotes the direct product of two graphs. iv. K mk 1 is itegral if ad oly if 4m+5 ad ( 1) 2 (4m + 1) + 4 are perfect squares. v. K p,q mk 1 is itegral if ad oly if pq ad pq(4m + 1) + 4 are perfect squares. vi. If G mk 1 is a itegral graph the (K 2 G) mk 1 is itegral, where deotes the direct product of two graphs. The above observatios eable us to costruct some class of ew itegral graphs.

13 SPECTRA OF GRAPH OPERATIONS BASED ON CORONA 67 Example 5.1. I the followig table we give ifiite ordered pairs (m,) for which the graph K mk 1 is itegral. m 6k + 3 (5k 2 + 5k + 1)(45k k 1), k = 1, 2, 6k 1 (5k 2 5k + 1)(45k 2 15k 1), k = 1, 2, 2k + 1 k 4 3k 2 + 1, k = 2, 3, 2 k 2 + 3k + 1, k = 1, 2, Example 5.2. a. K, 2 K 1 is itegral graph for all. b. K 1, 2 2 K 1 is itegral graph for all. Example 5.3. a. If = 2k ad m = k 2 k 1, the K mk 1 is itegral for k = 1, 2,. b. If = 2k 1 ad m = k 2 k 1, the K, mk 1 ad K 1, 2 mk 1 is itegral for k = 1, 2,. From Corollaries 4.1 ad 4.4 it follows that a. If G is a itegral r-regular graph (r 2) o vertices, the G mk 1 is itegral if ad oly if λ i (G) 2 i +4m(λ i(g)+r)+4 is a perfect square, for i = 1, 2,. b. If G is a itegral r-regular graph (r 2) o vertices, the G mk 1 is itegral if ad oly if 5λ 2 i (G) + 4m(λ i(g) + r) is a perfect square, for i = 1, 2,,. I particular, we have the followig i. K mk 1 ( 2) is itegral if ad oly if 4m( 2)+5 ad ( 1) 2 +8m( 1) + 4 are perfect squares. ii. K, mk 1 ( 2)is itegral if ad oly if m + 1, 2 + 8m + 4, are perfect squares. iii. K mk 1 ( 2) is itegral if ad oly if 4m( 2)+5 ad 5( 1) 2 +8m( 1) are perfect squares. iv. K, mk 1 ( 2) is ever a itegral graph for all ad m. The above observatios eables us to costruct some ew class of itegral graphs. Example 5.4. If m = k 2 k 1, the K 3 mk 1 is itegral for all k = 2, 3,. Now, we give methods to costruct ifiite family of cospectral graphs. From Theorem 3.1 ad 3.4 oe ca easily otice that a. If G 1 ad G 2 are adjacecy cospectral graphs ad H is a arbitrary graph, the i. G 1 H ad G 2 H are adjacecy cospectral. ii. G 1 H ad G 2 H are adjacecy cospectral. b. If G is a arbitrary graph ad H 1, H 2 are adjacecy cospectral graphs with Γ A(H1)(x) = Γ A(H2)(x), the i. G H 1 ad G H 2 are adjacecy cospectral. ii. G H 1 ad G H 2 are adjacecy cospectral.

14 68 CHANDRASHEKAR ADIGA AND RAKSHITH B. R. Similary usig Theorem 3.2, 3.5 ad 3.3, 3.6 oe ca costruct Laplacia cospectral ad sigless Laplacia cospectral graphs. Also from Theorem 4.1 ad 4.4 we have the followig results: a. If G 1 ad G 2 are adjacecy regular cospectral graphs ad H is a arbitrary graph, the i. G 1 H ad G 2 H are adjacecy cospectral. ii. G 1 H ad G 2 H are adjacecy cospectral. b. If G is a arbitrary regular graph ad H 1, H 2 are adjacecy cospectral graphs with Γ A(H1)(x) = Γ A(H2)(x), the i. G H 1 ad G H 2 are adjacecy cospectral. ii. G H 1 ad G H 2 are adjacecy cospectral. Similary usig Theorem 4.2, 4.5 ad 4.3, 4.6 oe ca costruct Laplacia cospectral ad sigless Laplacia cospectral graphs. Refereces [1] C. Adiga, R. Balakrisha, W. So, The skew eergy of a digraph, Liear Algebra Appl. 432 (7)(2010), [2] K. Baliska, D. Cvetković, Z. Radosavljević, S. Simić ad D. Stevaović, A survey o itegral graphs, Uiv. Beograd. Publ. Elektroteh. Fak. Ser. Mat. 13 (2002), [3] S. Barik, S. Pati ad B. K. Sarma, The spectrum of the coroa of two graphs, SIAM J. Discrete Math. 21 (1)(2007), [4] Cui S-Y, Tia G-X, The spectrum ad the sigless Laplacia spectrum of coroae. Liear. Algebra Appl., 437 (7) (2012), [5] D. Cvetković, New theorems for sigless Laplacia eigevalues, Bull. Acad. Serbe Sci. Arts, Cl. Sci. Math. Natur., Sci. Math., 137(33) (2008), [6] D. Cvetković, S.K. Simić, Towards a spectral theory of graphs based o the sigless Laplacia, II, Liear Algebra Appl., 432 (9) (2010), [7] D.M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs Theory ad Applicatios, third ed., Joha Ambrosius Barth, Heidelberg, [8] D.M. Cvetković, P. Rowliso, H. Simić, A Itroductio to the Theory of Graph Spectra, Cambridge Uiversity Press, Cambridge, [9] R. Frucht, F. Harary, O the coroa of two graphs, Aequatioes Math., 4 (3)(1970), [10] F. Harary, A. J. Schwek, Which Graphs have Itegral Spectra?, Graphs ad Combiatorics (R. Bari ad F. Harary, eds.), Spriger-Verlag, Berli (1974), [11] Y. Hou, W-C.Shiu, The spectrum of the edge coroa of two graphs, Electroic Joural of Liear Algebra, 20 (2010), Article 42, [12] G. Idulal, The spectrum of eighborhood coroa of graphs, Kragujevac J. Math., 35 (3) (2011), [13] G. Idulal, A. Vijayakumar, Some New Itegral Graphs, Applicable Aalysis ad Discrete Mathematics, 1 (2007), [14] J. La, B. Zhou, Spectra of graph operatios based o R-graph, Liear ad Multiliear Algebra, 63(7) (2015), [15] X.Li, Y.Shi, I.Gutma, Graph Eergy, Spriger, NewYork, [16] X. Liu, P. Lu, Spectra of subdivisio-vertex ad subdivisio-edge eighbourhood coroae, Liear Algebra Appl., 438 (8)(2013),

15 SPECTRA OF GRAPH OPERATIONS BASED ON CORONA 69 [17] X. Liu, S. Zhou, Spectra of the eighbourhood coroa of two graphs, Liear ad Multiliear Algebra, 62 (9) (2014), [18] C. McLema, E. EMcNicholas, Spectra of coroae, Liear Algebra Appl., 435 (5)(2011), [19] R. Merris, Laplacia matrices of graphs: a survey, Liear Algebra Appl., 197/198 (1994), [20] L. G. Wag, A survey of results o itegral trees ad itegral graphs, Uiversity of Twete, The Netherlads (2005). [21] L. G. Wag, H.J. Broersma, C. Hoede, X.Li, G. Still, Some families of itegral graphs, Discrete Math., 308 (24)(2008), Received by editors at ; Available olie Departmet of Studies i Mathematics, Uiversity of Mysore, Maasagagothri, Mysore , INDIA address: c adiga@hotmail.com, ramsc08@yahoo.co.i