Cospectrality of graphs
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1 Linear Algebra and its Applications 451 (2014) Contents lists available at ScienceDirect Linear Algebra and its Applications Cospectrality of graphs Alireza Abdollahi a,b,, Mohammad Reza Oboudi a,b a Department of Mathematics, University of Isfahan, Isfahan , Iran b School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box , Tehran, Iran article info abstract Article history: Received 17 October 2013 Accepted 15 February 2014 Available online xxxx Submitted by R. Brualdi MSC: 05C50 05C31 Keywords: Spectra of graphs Measures on spectra of graphs Adjacency matrix of a graph Richard Brualdi proposed in Stevanivić (2007) [6] the following problem: (Problem AWGS.4) Let G n and G n be two nonisomorphic graphs on n vertices with spectra λ 1 λ 2 λ n and λ 1 λ 2 λ n, respectively. Define the distance between the spectra of G n and G n as λ ( ( G n,g ) n ( n = λi λ ) 2 i or use i=1 ) n λi λ i. i=1 Define the cospectrality of G n by Let cs(g n )=min { λ ( G n,g n) : G n not isomorphic to G n }. cs n =max { cs(g n ): G n agraphonn vertices }. Problem A. Investigate cs(g n ) for special classes of graphs. Problem B. Find a good upper bound on cs n. In this paper we study Problem A and determine the cospectrality of certain graphs by the Euclidian distance. * Corresponding author. addresses: a.abdollahi@math.ui.ac.ir (A. Abdollahi), mr.oboudi@sci.ui.ac.ir (M.R. Oboudi) / 2014 Published by Elsevier Inc.
2 170 A. Abdollahi, M.R. Oboudi / Linear Algebra and its Applications 451 (2014) Let K n denote the complete graph on n vertices, nk 1 denote the null graph on n vertices and K 2 +(n 2)K 1 denote the disjoint union of the K 2 with n 2 isolated vertices, where n 2. In this paper we find cs(k n ), cs(nk 1 ), cs(k 2 + (n 2)K 1 )(n 2) and cs(k n,n ) Published by Elsevier Inc. 1. Introduction Throughout the paper all graphs are simple, that is finite and undirected without loops and multiple edges. By the spectrum of a graph G, we mean the multiset of eigenvalues of adjacency matrix of G. Richard Brualdi proposed in [6] the following problem: (Problem AWGS.4) Let G n and G n be two nonisomorphic graphs on n vertices with spectra λ 1 λ 2 λ n and λ 1 λ 2 λ n, respectively. Define the distance between the spectra of G n and G n as λ ( G n,g ) n ( ( n = λi λ ) 2 i or use i=1 Define the cospectrality of G n by ) n λi λ i. i=1 cs(g n )=min { λ ( G n,g n) : G n not isomorphic to G n }. Thus cs(g n ) = 0 if and only if G n has a cospectral mate. Let cs n =max { cs(g n ): G n a graph on n vertices }. This function measures how far apart the spectrum of a graph with n vertices can be from the spectrum of any other graph with n vertices. Problem A. Investigate cs(g n ) for special classes of graphs. Problem B. Find a good upper bound on cs n. In this paper we study Problem A and determine the cospectrality of certain graphs by the Euclidian distance, that is λ ( G n,g ) n ( n = λi λ ) 2. i i=1
3 A. Abdollahi, M.R. Oboudi / Linear Algebra and its Applications 451 (2014) For a graph G, V (G) ande(g) denote the vertex set and edge set of G, respectively; G denotes the complement of G and A(G) denotes the adjacency matrix of G. Fortwo graphs G and H with disjoint vertex sets, G + H denotes the graph with the vertex set V (G) V (H) and the edge set E(G) E(H), i.e. the disjoint union of two graphs G and H. The complete product (join) G H of graphs G and H is the graph obtained from G + H by joining every vertex of G with every vertex of H. Inparticular,nG denotes G + + G }{{} n and n G denotes G G G }{{}. n We denote by Spec(G) the multiset of the eigenvalues of the graph G. For positive integers n 1,...,n l, K n1,...,n l denotes the complete multipartite graph with l parts of sizes n 1,...,n l.letk n denote the complete graph on n vertices, nk 1 = K n denote the null graph on n vertices and P n denote the path with n vertices. By the previous notation, for any integer n>2, K 2 +(n 2)K 1 denotes the disjoint union of the K 2 with n 2 isolated vertices. In this paper we find cs(k n ), cs(nk 1 ), cs(k 2 +(n 2)K 1 ) (n 2) and cs(k n,n ). In particular, we find that there exists a unique graph G H such that λ(h, G H )=cs(h) ifh {K n,nk 1,K 2 +(n 2)K 1,K n,n }. The main results of our paper are the following: Theorem 1.1. For every integer n 2, cs(nk 1 )=2. In particular, λ(nk 1,G)=cs(nK 1 ) for some graph G if and only if G = K 2 +(n 2)K 1. Theorem 1.2. For every integer n 3, cs(k 2 +(n 2)K 1 )=2( 2 1) 2. Also, cs(k 2 )= λ(k 2, 2K 1 )=2. In particular, λ(k 2 +(n 2)K 1,G)=cs(K 2 +(n 2)K 1 ) for some graph G if and only if G = P 3 +(n 3)K 1. Theorem 1.3. For every integer n 2, cs(k n )=n 2 +n n n 2 +2n 7 2. In particular, λ(k n,g)=cs(k n ) for some graph G if and only if G = K n \ e, wherek n \ e is the graph obtaining from K n by deletion one edge e. Theorem 1.4. Let n 2 be an integer. Then cs(k n,n )=2(n n 2 1) 2. In particular, λ(k n,n,g)=cs(k n,n ) for some graph G if and only if G = K n 1,n Cospectrality of graphs with at most one edge In this section we will determine the cospectrality of graphs with at most one edge. Let G be a simple graph of order n and size m. Letλ 1 λ n be the eigenvalues of G. Itiswellknownthatλ λ n =0andλ λ n 2 =2m. Wenowgivethe proof of Theorem 1.1 in which we determine the cospectrality of graphs with no edge. Proof of Theorem 1.1. Let G n be a simple graph of order n and size m with eigenvalues λ 1 λ n. Since the eigenvalues of nk 1 are λ 1 = = λ n =0,then λ ( nk 1,G n) = λ λ n 2 =2m.
4 172 A. Abdollahi, M.R. Oboudi / Linear Algebra and its Applications 451 (2014) Since G n is not isomorphic to nk 1, m 1. So the minimum value of λ(nk 1,G n)is2 and it happens for G n = K 2 +(n 2)K 1. Proof of Theorem 1.2. Let n 3 be an integer and let G n be a simple graph of order n and size m with eigenvalues λ 1 λ n. Since the eigenvalues of K 2 +(n 2)K 1 are λ 1 =1,λ 2 = = λ n 1 =0,λ n = 1, λ ( K 2 +(n 2)K 1,G ) ( n = λ 1 1 ) 2 + λ λ 2 ( n 1 + λ n +1 ) 2 =2m +2 2λ 1 +2λ n. Now, we want to find the minimum value of m λ 1 + λ n among all graphs of order n and size m. By Perron Frobenius Theorem (see [2, Theorem 0.13]), λ n λ 1.Thus λ ( K 2 +(n 2)K 1,G n) =2m +2 2λ 1 +2λ n 2m 4λ We claim that for every graph G n K 2 +(n 2)K 1,P 3 +(n 3)K 1, the following holds Since λ ( K 2 +(n 2)K 1,G n) > 2( 2 1) 2. λ ( K 2 +(n 2)K 1,P 3 +(n 3)K 1 ) =2( 2 1) 2, the validity of our claim completes the proof. To prove the claim we consider the following cases: Case 1. Let λ Then (λ 1 1) Thus λ ( K 2 +(n 2)K 1,G ) n = ( λ 1 1 ) 2 + λ λ n ( λ n +1 ) > 2( 2 1) 2. Case 2. Let λ 1 < Then for m 4, 2m 4λ 1.Since λ ( K 2 +(n 2)K 1,G n) 2m 4λ 1 +2, it follows that λ(k 2 +(n 2)K 1,G n) 2 and so we are done. To complete the proof of our claim it remains to compute λ(k 2 +(n 2)K 1,G n) for all graphs G n with at most 3 edges. These graphs are as follows, nk 1, K 2 +(n 2)K 1,2K 2 +(n 4)K 1, P 3 +(n 3)K 1,3K 2 +(n 6)K 1, P 3 + K 2 +(n 5)K 1, K 3 +(n 3)K 1, P 4 +(n 4)K 1 and K 1,3 +(n 4)K 1. One can easily see that for the latter graphs the claim is valid. This completes the proof.
5 A. Abdollahi, M.R. Oboudi / Linear Algebra and its Applications 451 (2014) Cospectrality of the complete graph In this section we will determine the cospectrality of the complete graphs. Let G be a simple graph of order n and size m. In this section we show that for every integer n 2, cs(k n )=λ(k n,k n \ e), where e is an arbitrary edge of K n. First we prove some lemmas. Theorem 3.1. (Theorem of [3]) LetG be a graph of order n and H be an induced subgraph of G with order m. Suppose that λ 1 (G) λ n (G) and λ 1 (H) λ m (H) are the eigenvalues of G and H, respectively. Then for every i, 1 i m, λ i (G) λ i (H) λ n m+i (G). Let X be a graph. Recall that a partition π of V (X) with cells C 1,...,C r is equitable if the number of neighbors in C j of a vertex u in C i is a constant b ij, independent of u. Theorem 3.2. (Theorem and Exercise 3 in page 213 of [3]) If π is an equitable partition of a graph X, then the characteristic polynomial of A(X/π) divides the characteristic polynomial of A(X). Moreover, the spectral radius of A(X/π) is equal to the spectral radius of A(X). Note that in Theorem 3.2, X/π denotes the directed graph with the r cells of π as its vertices and b ij arcs from the ith to the jth cells of π is called the quotient of X over π, and denoted by X/π. The entries of the adjacency matrix of this quotient are given by A(X/π) ij = b ij. Let K t n be the graph obtained from K n by deleting n t 1 edges from one vertex of K n. By the following lemma one can compute the spectrum of K t n. Lemma 3.3. Let V (K n ) = {v 0,v 1,...,v n 1 }. Let 1 t n 2 and K t n = K n \ {v 0 v t+1,v 0 v t+2,...,v 0 v n 1 }. Suppose that f(λ) :=λ 3 (n 3)λ 2 (n+t 2)λ+t(n t 2). Then Spec ( Kn) t = { 1,..., 1,λ }{{} 1,λ 2,λ 3 }, n 3 where λ 1, λ 2, λ 3 are the roots of the polynomial f(λ) and 1 max{λ 1,λ 2,λ 3 }. Proof. Let A = {v 0 }, B = {v 1,...,v t } and C = {v t+1,...,v n 1 }. It is easy to see that the partition {A, B, C} of vertices of Kn t is an equitable partition with the following matrix 0 t 0 M = 1 t 1 n 1 t, 0 t n t 2
6 174 A. Abdollahi, M.R. Oboudi / Linear Algebra and its Applications 451 (2014) where the first (second and third) row and column corresponds to A (B and C, respectively). Since Spec(K n 1 )={n 2, 1,..., 1} and K }{{} n 1 is an induced subgraph of Kn, t n 2 by interlacing Theorem 3.1 we conclude that Kn t has at least n 3 eigenvalues 1. By Theorem 3.2, the three remaining eigenvalues of Kn t are the eigenvalues of the matrix M and the largest eigenvalue of Kn t is among the latter three eigenvalues. On the other hand f(λ) =det(λi M) andf( 1) 0. This completes the proof. Corollary 3.4. For every integer n 2 and every arbitrary edge e of K n, { n 3+ n2 +2n 7 Spec(K n \ e) =, 0, 1,..., 1, n 3 } n 2 +2n 7. 2 }{{} 2 n 3 Proof. By the notation of Lemma 3.3, K n \ e = K n 2 n. This implies the result. Remark 3.5. Let G be a graph of order n and size m. Letλ 1 λ n be the eigenvalues of G. Sincen are the eigenvalues of K n, one can obtain that λ(k n,g)=2m 2nλ 1 + n 2 n. This equality shows that to obtain cs(k n )itis sufficient to obtain a graph G for which the parameter m nλ 1 has the minimum value. In sequel we show that among all graphs G of order n except K n, the minimum value of m nλ 1 (G) is attained on the graph K n \ e. Lemma 3.6. For every integer n 3 and every edge e of K n, λ(k n,k n \e) < 2. Ifn =2, then λ(k n,k n \ e) =2. Proof. For n = 2, there is nothing to prove. Let n 3. Using Corollary 3.4 and Remark 3.5 one can obtain the result. Lemma 3.7. Let n 3 be an integer. Let K = K n \{e, e },wheree and e are two adjacent edges in K n.thenλ(k n,k) >λ(k n,k n \ e). Proof. Note that K = Kn n 3.ByLemma 3.3 λ 1 (K) is a root of the polynomial f(λ) = λ 3 (n 3)λ 2 (2n 5)λ + n 3. Since the roots of the derivation f (λ) (with respect to λ) off(λ) are n 3± n 2 6 3, f is an increasing function on the interval [ n 3+ n 2 6 3, ). Using Corollary 3.4 one can see that λ 1 (K n \ e) 1 n > n 2 > n 3+ n Itis not hard to see that f(λ 1 (K n \ e) 1 n ) > 0. On the other hand f is increasing on [ n 3+ n 2 6 3, ). Thus f(λ) > 0 for every λ λ 1 (K n \ e) 1 n.sinceλ 1(K) isaroot of f(λ), we conclude that λ 1 (K) < λ 1 (K n \ e) 1 n(n 1) n. This shows that 2 2 nλ 1 (K) > n(n 1) 2 1 nλ 1 (K n \ e). Equivalently, λ(k n,k) > λ(k n,k n \ e) (see Remark 3.5).
7 A. Abdollahi, M.R. Oboudi / Linear Algebra and its Applications 451 (2014) We need the following theorems to prove the main result of this section. Theorem 3.8. (See [5], and also Theorem 6.7 of [2].) A graph has exactly one positive eigenvalue if and only if its non-isolated vertices form a complete multipartite graph. Lemma 3.9. Let G be a graph with eigenvalues λ 1 λ 2 λ 3 λ n. The graph G is isomorphic to one the following graphs if and only if λ 1 > 0, λ 2 0 and λ 3 < 0. (1) G = K n, (2) G = K 1 + K n 1, (3) G = K 2,1,...,1 = K n \ e for an edge e of K n. Proof. Suppose that λ 1 > 0, λ 2 0andλ 3 < 0. By Theorem 3.8, there exist some integers t 1, n 1,...,n t 1andr 0such that G = rk 1 + K n1,...,n t.sinceλ 3 < 0, r 1. Thus it is enough to investigate the cases G = K n1,...,n t and G = K 1 + K n1,...,n t. Note that t 2, otherwise G = K n, a contradiction. Suppose that G = K 1 +K n1,...,n t.ifn i 2forsomei, theng has a 3K 1 as an induced subgraph and so it follows from Interlacing Theorem that λ 3 0, a contradiction. Therefore G = K 1 + K n 1 in this case. Now assume that G = K n1,...,n t.ifn 1 = = n t =1,thenG = K n.thuswemay assume that n i 2forsomei. Suppose that n i,n j 2 for some distinct i and j. Thus the cycle C 4 of length 4 is an induced subgraph of G. Since the eigenvalues of C 4 are 2, 0, 0, 2, it follows from Interlacing Theorem that λ 3 0, a contradiction. Thus we can assume that n 1 2andn 2 = = n t =1.Ifn 1 =2,thenG = K 2,1,...,1 = K n \ e for some edge e. Therefore we may assume n 1 3. This shows that the star K 1,3 is an induced subgraph of G. Since the eigenvalues of K 1,3 are 3, 0, 0, 3, it follows from Interlacing Theorem 3.1 that λ 3 0, a contradiction. The converse follows from Corollary 3.4 and the fact that the eigenvalues of the complete graph K n are n 1, 1,..., 1. Theorem (See [4].) Let G be a graph without isolated vertices and let λ 2 (G) be the second largest eigenvalue of G. Then0 <λ 2 (G) 2 1 if and only if one of the following holds: (1) G = ( t (K 1 + K 2 )) K n1,...,n m. (2) G = (K 1 + K r,s ) K q. (3) G = (K 1 + K r,s ) K p,q. Now, we are in a position to prove the main result of this section. Theorem Let G be a graph of order n. IfG K n and G K n \e, thenλ(k n,g) 2 or λ(k n,g) >λ(k n,k n \ e).
8 176 A. Abdollahi, M.R. Oboudi / Linear Algebra and its Applications 451 (2014) Proof. Since G K n,k n \e, n 3. Let λ 1 λ n be the eigenvalues of G. Therefore λ(k n,g)= ( λ 1 (n 1) ) 2 +(λ2 +1) 2 + +(λ n +1) 2. It is easy to see that if one of the following cases holds, then λ(k n,g) 2. Case 1. λ 1 (n 1) 2. Case 2. λ 2 λ 3 0. Case 3. λ Now, suppose that none of the above cases occurs. Therefore we may assume that λ 1 >n 1 2, λ 2 < 2 1andλ 3 < 0. Suppose that λ 2 0. Thus it follows from Lemma 3.9 and the hypothesis, G = K 1 + K n 1. Therefore λ(k n,g)=2. Now suppose that λ 2 > 0. Thus λ 1 >n 1 2, 0 <λ 2 < 2 1andλ 3 < 0. Hence Theorem 3.10 can be applied. Case a. G = ( t (K 1 + K 2 )) K n1,...,n m.ift =0,thenG = K n1,...,n m, which is not possible by Theorem 3.8. Ift 2, then (K 1 + K 2 ) (K 1 + K 2 ) is an induced subgraph of G. Since Spec ( (K 1 + K 2 ) (K 1 + K 2 ) ) = { ,.41421,.26795, 1, 1, }, by Interlacing Theorem λ 3 0, a contradiction. Now, suppose that t = 1. If there exists i such that n i 3, K 1,3 is an induced subgraph of G. Now Interlacing Theorem implies that λ 3 0, a contradiction. Now, we may assume that n i 2 for all i. Ifm =1andn 1 =1,G = (K 1 + K 2 ) K 1, then Spec(G) ={ ,.31111, 1, 1, } and λ(k 4,G) > 2. If m =1andn 1 =2,G =(K 1 + K 2 ) K 2,then Spec(G) ={ ,.32164, 0, 1, }. Hence λ 3 0, a contradiction. Thus we may assume that m 2. If there exist i and j such that n i,n j 2, then C 4 is an induced subgraph of G, and since Spec(C 4 )= {2, 0, 0, 2}, Interlacing Theorem implies that λ 3 0, a contradiction. Thus we can assume that G = (K 1 + K 2 ) K n 3 or G = (K 1 + K 2 ) K 2,1...,1.WemayalsowriteG as follows: (1) G = (K 1 + K 2 ) K n 3 = K n \{e, e } or (2) G = (K 1 + K 2 ) (K n 3 \ e )=K n \{e, e,e }
9 A. Abdollahi, M.R. Oboudi / Linear Algebra and its Applications 451 (2014) where e, e,e E(K n ), e and e have a common vertex and e, e and e,e are pairwise non-adjacent in K n. If (1) happens then it follows from Lemma 3.7 that λ(k n,g) > λ(k n,k n \ e). Now suppose that (2) happens. Assume e = {v 1,v 2 }, e = {v 1,v 3 } and e = {v 4,v 5 }, where v 1,...,v 5 V (K n ). The induced subgraph on vertices v 1,...,v 5 is (K 1 + K 2 ) K 2.Since Spec ( (K 1 + K 2 ) K 2 ) = { ,.32164, 0, 1, }, it follows from Interlacing Theorem that λ 3 0, a contradiction. Now it remains to investigate the following cases: Case b. G = (K 1 + K r,s ) K q. Case c. G = (K 1 + K r,s ) K p,q. In both cases b and c, if either of r or s is greater than 1, then an induced subgraph isomorphic to K 1 + K 1,2 in G occurs. It now follows from Interlacing Theorem that λ 3 0asSpec(K 1 + K 1,2 )={ 2, 0, 0, 2}. Now we may assume that r = s =1. In both cases b and c, if either of q or p is greater than 1, then an induced subgraph isomorphic to H =(K 1 + K 2 ) K 2 in G occurs. Since Spec(H) ={ , 1, 0,.32164, } it follows from Interlacing Theorem that λ 3 0, a contradiction. Therefore q = p =1 and so G is isomorphic to one the following graphs: (i) G = G 1 =(K 1 + K 2 ) K 1, (ii) G = G 2 =(K 1 + K 2 ) K 2. Since and Spec(G 1 )={ , 1,.31111, } Spec(G 2 )={ , 1, 1,.35793, }, λ(k 4,G 1 ) > 2andλ(K 5,G 2 ) > 2. This completes the proof. Proof of Theorem 1.3. By Lemmas 3.6 and 3.7 and Theorem 3.11, it follows that cs(k n )=λ(k n,k n \ e) =n 2 + n n n 2 +2n 7 2. This completes the proof. 4. Cospectrality of complete bipartite graphs Let K m,n be the complete bipartite graph with parts of sizes m and n. Itisknownthat if G is a cospectral mate of K m,n,theng is bipartite and it follows from Theorem 3.8 that G = K r,s + tk 1 for some positive integers r and s andanintegert 0, where
10 178 A. Abdollahi, M.R. Oboudi / Linear Algebra and its Applications 451 (2014) r + s + t = n + m and rs = mn. Therefore if m and n are positive integers such that there exist integers r > 0ands>0 satisfying rs = mn and r + s<m+ n, then cs(k m,n )=0. Proposition 4.1. Let m and n be positive integers. Then cs(k m,n ) > 0 if and only if the minimum of x + y for all positive integers x, y such that xy = mn is attained on {x, y} = {m, n}. If n>0, then cs(k n,n ) > 0byProposition 4.1. We now compute the value of cs(k n,n ). We need the following result in the latter computation. Theorem 4.2. (Theorem 2 of [1]) LetG be a graph of order n without isolated vertices. Then 0 <λ 2 (G) < 1 3 if and only if G = (K 1 + K 2 ) K n 3,whereλ 2 (G) is the second largest eigenvalue of G. Proof of Theorem 1.4. Since for every positive integer m and n Spec(K m,n )={ mn, 0,...,0, mn}, }{{} m+n 2 it follows that λ(k n,n,k n 1,n+1 )=2(n n 2 1) 2.Ifn = 2, then the result follows from Fig. 3, where the cospectrality of all graphs of order 4 is computed. Now assume that n 3. Suppose, for a contradiction, that G is a graph non-isomorphic to K n,n and K n 1,n+1 such that λ(k n,n,g) < 2(n n 2 1) 2. Assume λ 1 λ 2n are the eigenvalues of G. Sinceforn 3, 2(n n 2 1) 2 < 1 9 we obtain λ 2 < 1 3.Sincefor every graph except of the complete graph, the second largest eigenvalue is non-negative (see [1, part 3 of Theorem 1]), the latter inequality shows that 0 λ 2 < 1 3.Wecan distinguish the following cases: Case 1. Suppose that λ 2 =0.ThusbyTheorem 3.8, there exist some positive integers k and n 1,...,n k and an integer t 0 such that G = tk 1 + K n1,...,n k.ifk =1,then G = K 2n and so λ(k n,n,g)=2n 2 18, a contradiction. If k =2,thenG = tk 1 + K r,s for some r and s such that r +s =2n t. Inthiscasewehaveλ(K n,n,g)=2(n rs) 2. It is not hard to see that if {r, s} {n, n} and {r, s} {n 1,n+1}, then2(n rs) 2 > 2(n n 2 1) 2. Therefore k 3. If n 1 = = n k =1,thenG = tk 1 + K 2n t and so λ(k n,n,g) > 2(n n 2 1) 2, a contradiction. Now we may assume that G has K 1,1,2 as an induced subgraph. Since λ 3 (K 1,1,2 )= 1, then it follows from Interlacing Theorem that λ 2 2n 1 1andsoλ(K n,n,g) 1, a contradiction. Case 2. Assume that 0 <λ 2 < 1 3.ByTheorem 4.2, we conclude that there exists an integer t 0 such that G = tk 1 +(K 1 + K 2 ) K 2n t 3.If2n t 3 = 1, then it is easy to λ(k n,n,g) > 2(n n 2 1) 2, a contradiction. If 2n t 3 > 1, then G has K 1,1,2 as an induced subgraph and the rest is similar to previous part. This completes the proof.
11 A. Abdollahi, M.R. Oboudi / Linear Algebra and its Applications 451 (2014) Fig. 1. Cospectrality of graphs with 2 vertices. Fig. 2. Cospectrality of graphs with 3 vertices. 5. Cospectrality of graphs of order at most 4 In this section (see Figs. 1, 2 and 3), we find the cospectrality of all graphs of order at most 4. Based on cospectrality of graphs with at most 4 vertices, one can observe the following facts: (1) It is not in general true that if cs(g) =λ(g, H) for some graph H, thencs(h) = λ(g, H); for example cs(a 4 )=λ(a 4,A 3 ) but cs(a 3 )=λ(a 3,A 2 )(seefig. 2); also cs(b 3 )=λ(b 3,B 4 ) but cs(b 4 )=λ(b 4,B 9 )(seefig. 3). (2) It is not in general true that if G is a regular graph and cs(g) =λ(g, H) forsome graph H, thenh is also regular; for example cs(a 4 )=λ(a 4,A 3 )(seefig. 2); also cs(b 9 )=λ(b 9,B 4 )(seefig. 3). The following question is natural to ask. Question 5.1. Let G and H be two graphs such that λ(g, H) =cs(g) =cs(h). For which graph theoretical property ρ, ifg has ρ then so does H? It is well-known that if λ(g, H) =cs(g) =cs(h) = 0 (that is Spec(G) =Spec(H)), then the answer of Question 5.1 for graph properties such as being bipartite and regularity is positive. We propose the following question to finish the paper. Question 5.2. Is there a constant c for which if cs(g) = λ(g, H), then E(G) E(H) c?
12 180 A. Abdollahi, M.R. Oboudi / Linear Algebra and its Applications 451 (2014) Fig. 3. Cospectrality of graphs with 4 vertices. Remark 5.3. Concerning Question 5.2 we have the following remarks. By Figs. 2 and 3, c can be chosen 1 for graphs with at most 4 vertices. However for graphs of order 5, it is easy to see that cs(k 2,3 )=2( 6 2) 2 and λ(k 2,3,G)=2( 6 2) 2 for some graph G if and only if G = K 1,4 or G = K 2,2 + K 1. By similar arguments given in Section 4, it is not hard to see that for all n 3, cs(k n,n+1 )=λ(k n,n+1,k n 1,n+2 )= 2( n 2 + n n 2 + n 2) 2 and the graph K n 1,n+2 is the only graph G satisfying the equality λ(k n,n+1,g) = cs(k n,n+1 ). Therefore for graphs with at least 2n +1 7 vertices if the constant c exists it is at least 2. Acknowledgements We are grateful to referees for their useful comments. The research of the first author was in part supported by a grant (No ) from School of Mathematics, Institute for Research in Fundamental Sciences (IPM). The research of the second author was
13 A. Abdollahi, M.R. Oboudi / Linear Algebra and its Applications 451 (2014) in part supported by a grant (No ) from School of Mathematics, Institute for Research in Fundamental Sciences (IPM). The research of the authors is financially supported by the Center of Excellence for Mathematics, University of Isfahan. References [1] D. Cao, H. Yuan, Graphs characterized by the second eigenvalue, J. Graph Theory 17 (3) (1993) [2] D.M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs, Theory and Application, Academic Press, Inc., New York, [3] C. Godsil, G. Royle, Algebraic Graph Theory, Springer, New York, [4] M. Petrović, On graphs whose second largest eigenvalue does not exceed 2 1, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 4 (1993) [5] J.H. Smith, Symmetry and multiple eigenvalues of graphs, Glas. Mat. Ser. III 12 (1) (1977) 3 8. [6] D. Stevanivić, Research problems from the Aveiro Workshop on Graph Spectra, Linear Algebra Appl. 423 (2007)
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