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Seidel-Estrada Index Jalal Askari1, Ali Iranmanesh1* and Kinkar Ch Das2

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View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Crossref Askari et al. Journal of Inequalities and Applications (2016)2016:120 DOI 10.1186/s13660-016-1061-9 R E S E A R C H Open Access Seidel-Estrada index Jalal Askari1, Ali Iranmanesh1* and Kinkar Ch Das2 *Correspondence: [email protected] Abstract 1Department of Mathematics, Faculty of Mathematical Sciences, Let G be a simple graph with n vertices and (0, 1)-adjacency matrix A.Asusual, Tarbiat Modares University, P.O. Box S(G)=J –2A – I denotes the Seidel matrix of the graph G. Suppose θ1, θ2, ..., θn and 14115-137, Tehran, Iran λ1, λ2, ..., λn are the eigenvalues of the adjacency matrix and the Seidel matrix of G, Full list of author information is n θi available at the end of the article respectively. The Estrada index of the graph G is defined as i=1 e .Wedefineand n λi investigate the Seidel-Estrada index, SEE = SEE(G)= i=1 e . In this paper the basic properties of the Seidel-Estrada index are investigated. Moreover, some lower and upper bounds for the Seidel-Estrada index in terms of the number of vertices are obtained. In addition, some relations between SEE and the Seidel energy Es(G)are presented. MSC: 05C50; 05C90 Keywords: eigenvalue; Seidel matrix; Seidel-Estrada index 1 Introduction Throughout this paper, let G be a simple graph with vertex set V = {v, v,...,vn}.The adjacency matrix A(G)=[aij]ofG isabinarymatrixofordern such that aij = if the vertex vi is adjacent to the vertex vj, and otherwise. The Seidel matrix S(G)=[sij]isequalto Jn –A(G)–In,wherethesymbolJn denotes the square matrix of order n all of whose entries are equal to . Since A(G)andS(G) are real symmetric matrices, their eigenvalues must be real. The eigenvalues of G are referred to as the eigenvalues of A(G), denoted by θ(A(G)), θ(A(G)),...,θn(A(G)) and similarly, λ(S(G)) ≥ λ(S(G)) ≥···≥λn(S(G)), the Seidel eigenvalues of G.Forsimplicity,wewriteλi instead of λi(S(G)). The sequence of n Seidel eigenvalues is called the Seidel spectrum of G (for short S-spec(G)). We now present an example of pairs of graphs on n vertices with the same Seidel spectrum such that one of them is a connected graph and the other one is not. Example Here we address two examples from non-isomorphic graphs which are co- spectral: (i) S-spec(Kp,q)=S-spec(K n)ifp + q = n, (ii) S-spec(Kn/ ∪ Kn/)=S-spec(Kn)(n is even). In our recent studies on Seidel eigenvalues it has been shown that a lower and upper bound exists for the sum of powers of the absolute eigenvalues of the Seidel matrix, sug- gesting a common core architecture similar to the cases of adjacency and signless Lapla- cian matrix []. The reader can find more information related to the eigenvalues of the © 2016 Askari et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Askari et al. Journal of Inequalities and Applications (2016)2016:120 Page 2 of 9 adjacency matrix and the spectrum of G in []. The Estrada index of a graph G is defined as n EE(G)= eθi(A(G)).() i= This graph-spectrum-based structural descriptor was first proposed by Estrada in ; see [–]. Already, de la Peña et al. [] proposed to call it the Estrada index, a name that in the meantime has been commonly accepted. Several kinds of Estrada indices were dis- cussed in [–] and the references therein. For the recent work of the mathematical prop- erties on the Estrada and signless Laplacian Estrada indices, see [, ]. In this review, we summarize some indirect evidence to support the concept of a Seidel matrix. Similarly, we define the Seidel-Estrada index for the graph G in full analogy with equation ()as n SEE(G)= eλi .() i= For details on the theory of the Estrada index and several lower and upper bounds, see [, , , ]. Ayyaswamy et al. [] gave a lower bound for a signless Laplacian of the graph using the numbers of vertices and edges. A conference matrix is a square matrix C of order n with zero diagonal and ± off the diagonal, such that CCT =(n –)I.IfC is symmetric, then C is the Seidel matrix of a graph and this graph is called a conference graph;see[, ]. The aim of this paper is to find the upper and lower bounds for the Seidel-Estrada index of the graph G. The rest of the paper is organized as follows: In Section ,wegive some definitions and obtain some upper and lower bounds for the Seidel-Estrada index. In Section , we present a relation between the Seidel-Estrada index and the Seidel energy of a graph G, and we prove several results on the Seidel-Estrada index. 2 Estimates of the Seidel-Estrada index Here we give some new lower and upper bounds on Seidel-Estrada index. For convenience, we give some notation and properties which will be used in the following proofs of our n k k n | |k x results. Let Sk = Sk(G)= i=(λi) ,andS (G)= i= λi .FromtheTaylorexpansionofe , it is easy to see that the Seidel-Estrada index and Sk(G)ofG are related by ∞ S (G) SEE(G)= k .() k! k= It is easy to see that any graph G of order n ≥ hasSEE(G)>n. (If equality holds, then λ = λ = ···= λn = . By Lemma .(ii), we can get a contradiction.) Lemma . [] For any graph G with n vertices, we have n (i) S(G)= λi = trace S(G) =, i= n (ii) S (G)=S(G)= λi = trace S (G) =(n –) +(n –)=n(n –), i= Askari et al. Journal of Inequalities and Applications (2016)2016:120 Page 3 of 9 n ≤ ≤ (iii) S(G)= λi S (G) (n –) +(n –), i= n (iv) S (G)= |λi| ≥ n (n –) , i= n k k k (v) Sk(G) ≤ S (G)= |λi| ≤ (n –) +(n –), k =,,..., i= n k k k (vi) S (G)= |λi| ≥ n (n –) , k =,,.... i= Lemma . [] Let B be an n × n symmetric matrix with eigenvalues λ ≥ λ ≥···≥λn and let Bk be its leading k × ksubmatrixofB. Then, for i =,,...,k, λn–i+(B) ≤ λk–i+(Bk) ≤ λk–i+(B), () where λi(B) is the ith greatest eigenvalue of B. Lemma . Let G be a graph of order n ≥ . Then λ ≥ . Proof Since n ≥ , therefore K or K must be an induced subgraph of G.Sinceλ(K)= λ(K ) = , by Lemma ., we get the required result. Theorem . Let G be a simple graph with n ≥ and det S(G) =. Then the Seidel- Estrada index of G is bounded by √ n(n –)<SEE(G)<n –+e n(n–).() Proof (a) To prove this theorem, we apply a technique similar to the proof of Theorem in []. At first we prove that the left inequality of (): From (), we get n SEE(G)= eλi + eλi eλj .() i= i<j In view by the inequality between the geometric and arithmetic mean, we get n n– n(n–) n(n–) eλi eλj ≥ n(n –) eλi eλj = n(n –) eλi i<j i<j i= = n(n –) eS(G) n = n(n – ). () By using the power series expansion, and Lemma .,weget ∞ n n k n n n n k (λi) (λi) (λi) (λi) (λi) eλi = = + + + k! ! ! ! k! i= i= k= i= i= i= i= k≥ n (λ )k n (λ )k = S +S +S + i = n ++n(n –)+ i . k! k! i= k≥ i= k≥ Askari et al. Journal of Inequalities and Applications (2016)2016:120 Page 4 of 9 k k (λi) ≥ (λi) ∈ Since k≥ k! k≥ k! , we shall use a multiplier γ [, ], so as to arrive at n n k (λi) eλi ≥ n +n(n –)+γ k! i= i= k≥ n (λ )k = n +n(n –)–γ n – γ n(n –)+γ i k! i= k≥ = n +n(n –)–γ n – γ n(n –)+γ SEE(G). () By substituting ()and()backinto()andsolvingforSEE(G), we obtain γ γ SEE(G) ≥ + + n( – γ /) – n( + γ /). () Now, we consider a function x x x x f (x)= + + n – – n + .() We have f (x)<forx ≥ . Thus f (x) is a monotonically decreasing function for x >. Consequently, the best lower bound for SEE(G)isattainedγ = . Setting γ =in(), we arrive at the first half of Theorem .: SEE(G) ≥ n(n –). Now, we have to prove that the lower bound is strict. For this purpose, we assume that the left equality holds in (). Then we have eλi+λj = eλk +λ , for any i, j, k, ∈{,,...,n}, that is, eλ+λ = eλ+λ = ···= eλ+λn = eλ+λ , and hence λ = λ = ···= λn. By Lemma . and the trace of S(G), we can get a contradiction. Thus the left equality in ()isstrict. (b) Let us prove now the right inequality. Since f (x)=ex monotonically increases in the interval (–∞, ∞), we starting with equa- tion (), we get ∞ n (λ )k n (|λ |)k SEE(G)=n + i ≤ n + i k! k! i= k= i= k≥ n k [(λ )] = n + i k! k≥ i= Askari et al. Journal of Inequalities and Applications (2016)2016:120 Page 5 of 9 √ k ( n(n –))k ≤ n + S (G) = n + k! k! k≥ k≥ ∞ √ ( n(n –))k √ = n –+ = n –+e n(n–).() k! k= Suppose that the right equality holds in ().
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