Estrada Index of Random Bipartite Graphs

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1 Symmetry 25, 7, ; doi:.339/sym74295 OPEN ACCESS symmetry ISSN Article Estrada Index of Random Bipartite Graphs Yilun Shang Department of Mathematics, Tongji University, Shanghai 292, China; Academic Editor: Sergei D. Odintsov Received: 7 October 25 / Accepted: 3 November 25 / Published: 7 December 25 Abstract: The Estrada index of a graph G of n vertices is defined by EEG n i eλ i, where λ, λ 2,, λ n are the eigenvalues of G. In this paper, we give upper and lower bounds of EEG for almost all bipartite graphs by investigating the upper and lower bounds of the spectrum of random matrices. We also formulate an exact estimate of EEG for almost all balanced bipartite graphs. Keywords: Estrada index; random graph; eigenvalues; it spectral distribution. Introduction The topological structures of many social, biological, and technological systems can be characterized by the connectivity properties of the interaction pathways edges between system components vertices []. Starting with the Königsberg seven-bridge problem in 736, graphs with bidirectional or symmetric edges have ideally epitomized structures of various complex systems, and have developed into one of the mainstays of the modern discrete mathematics and network theory. Formally, a simple graph G consists of a vertex set V {, 2,, n} and an edge set E V V. The adjacency matrix of G is a symmetric, -matrix AG a ij R n n, where a ij a ji if vertices i and j are adjacent, and a ij a ji otherwise. It is well-known in algebraic graph theory that AG has exactly n real eigenvalues λ λ 2 λ n due to its symmetry. They are usually called the spectrum eigenvalues of G itself [2]. A spectral graph invariant, the Estrada index EEG of G, is defined as EEG n e λ i. This quantity was introduced by Estrada [3] in 2. It has noteworthy chemical applications, such as quantifying the degree of folding of long-chain molecules and the Shannon entropy [3 7]. The Estrada i

2 Symmetry 25, index provides a remarkable measure of subgraph centrality as well as fault tolerance in the study of complex networks [,8 ]. Building upon varied symmetric features in graphs, mathematical properties of this invariant can be found in, e.g., [ 8]. The Estrada index can be readily calculated once the eigenvalues are known. However, it is notoriously difficult to compute the eigenvalues of a large matrix even for, -matrix AG. In the past few years, researchers managed to establish a number of lower and upper bounds to estimate this invariant see [3] for an updated survey. A common drawback is that only few classes of graphs attain the equalities of those bounds. Therefore, one may naturally wonder the typical behavior of the invariant EEG for most graphs with respect to other graph parameters such as the number of vertices n. The classical Erdős Rényi random graph model G n p includes the edges between all pairs of vertices independently at random with probability p [9]. It has symmetric, bell-shaped degree distribution, which is shared by many other random graph models. Regarding the Estrada index, Chen et al. [2] showed the following result: Let G n p G n p be a random graph with a constant p,, then EEG n p e np e O n + o Here, we say that a certain property P holds in G n p almost surely if the probability that a random graph G n p has the property P converges to as n tends to infinity. Therefore, the result presents an analytical estimate of the Estrada index for almost all graphs. Our motivation in this paper is to investigate the Estrada index of random bipartite graphs, which is a natural bipartite version of Erdős Rényi random graphs. Bipartite graphs appear in a range of applications in timetabling, communication networks and computer science, where components of the systems are endowed with two different attributes and symmetric relations are only established between these two parts [2 23]. Formally, a bipartite graph is a graph whose vertices can be divided into two disjoint sets V and V 2 such that every edge connects a vertex in V to a vertex in V 2. A bipartite graph is a graph that does not contain any odd-length cycles; chemical trees are bipartite graphs. The random bipartite graph model is denoted by G n,n 2 p, where n i V i for i, 2, satisfying n + n 2 n. The authors in [2] posed the following conjecture pertaining to bipartite graphs. Conjecture. Let G n,n 2 p G n,n 2 p be a random bipartite graph with a constant p,. Then if and only if n 2 /n. EEG n,n 2 p e n 2 p e O n + o 2 In this paper, by means of the symmetry in G n,n 2 p and the spectral distribution of random matrix, we obtain lower and upper bounds for EEG n,n 2 p. For ease of analysis, we assume that n 2 /n : y, ]. We establish the estimate Theorem. e n 2p e O n + o EEG n,n 2 p e n p e O n + O 3 Thus a weak version of Conjecture follows readily: EEG n,n 2 p e n 2 p e O n + O

3 Symmetry 25, holds, provided n 2 /n i.e., y. 2. Examples In this section, we give some analytical and numerical examples to demonstrate that our estimate 3 is valid, and is better than some existing bounds obtained by algebraic approaches. Example. Let G be a bipartite graph with vertex set V V V 2, V V + V 2 n + n 2 n and m edges. In [] and [8], it was proved that n2 + 4m EEG n cosh m, 4 where the equality on the left-hand side of 4 holds if and only if G is an empty graph, while the equality on the right-hand side of 4 holds if and only if G is a union of a complete bipartite graph and some isolated vertices. If G G n,n 2 p, then the number of edges is m n n 2 p [9]. Hence, the lower bounds in 4 becomes n2 + 4m n 2 + 4n n 2 p e n 2p e O n + o The upper bound in 4 becomes if n 2 /n y > p. n cosh m n 2 + e n n 2 p + e n n 2 p e np e n yp p + o e n p e n yp p +y + o e n p e O n + O, Example 2. In [5], the authors bounded the Estrada index of a graph G of order n by its graph energy EG: EEG n + e EG 2, with equality if and only if EG. If G G n,n 2 p, it can be shown that [24] EG 2n 2 n Cpn 3/2 + on 3/2, where Cp > is a constant depending on p. Hence, we have We calculate the upper bound in 5 as 2n EEG n + e 2 n Cpn 3 2 +o n 3 2, 5 2n n + e 2 n Cpn 3 2 +o n 2 3 n 3 2 n p + o e n p e n p e 2yCpn 3 2 n 3 2 +o 2y+y e 2 3 Cp+o n 3 2 np+y + o e n p e O n + O.

4 Symmetry 25, Example 3. Let G be a bipartite graph with n vertices and m edges. Denote by n the number of zero eigenvalues in the spectrum of G. It was proved in [25] and [2] that 2m EEG n + n n cosh, 6 n n where the equality holds if and only if n n is even, and G consists of copies of complete bipartite graphs K ai,b i, i,, n n /2, such that all products a i b i are mutually equal. If G G n,n 2 p, then the number of edges is m n n 2 p Hence, the lower bound in 6 becomes 2m n + n n cosh n n n + 2 n n e n e n 2p e n 2p 2n n 2 p n n 2 e 2n n 2 p n n 2 p + o n 2 e 2ynp ynp +y + o e n 2p e O n + o. Example 4. An upper bound for EEG n,n 2 p was derived in [2] as Since + e 2n n 2 p n n EEG n,n 2 p e np e O n + o 7 e n p e O n + O e np e O n + o ynp + oe +y o, our upper bound in Theorem is better than that in 7. In Table, we compare the estimates for EEG n,n 2 /2 in Theorem with numerical value obtained by Matlab software. The results are in good agreement with the theory. Table. Estrada index of G n,n 2 p with n 3 and p /2. Numerical results are based on average over independent runs. n 2 y Theoretical lower bound Numerical value Theoretical upper bound 3 e 5 e O o e e 5 e O O 27.9 e 35 e O o e e 5 e O O 24.8 e 2 e O o e e 5 e O O 2.7 e 5 e O7.4 + o e e 5 e O7.4 + O 3. Proof of Theorem This section is devoted to the proof of Theorem, which heavily relies on the symmetry in G n,n 2 p. Throughout the paper, we shall understand p, as a constant. Let M m ij R n n be a random matrix, where the entries m ij i j are independent and identically distributed with

5 Symmetry 25, m ij m ji. We denote by the eigenvalues of M by λ M, λ 2 M,, λ n M and its empirical spectral distribution by Φ M x n #{λ im λ i M x, i, 2,, n} n {λi M x}. n i Lemma. Marčenko Pastur Law [26] Let X x ij R n 2 n be a random matrix, where the entries x ij are independent and identically distributed with mean zero and variance p p. Suppose that n, n 2 are functions of n, and n 2 /n y,. Then, with probability, the empirical spectral distribution Φ XX x converges weakly to the Marčenko Pastur Law F n T y as n, where F y has the density f y x b xx a{a x b} 2πp pxy and has a point mass /y at the origin if y >, where a p p y 2 and b p p + y 2. The above result formulates the it spectral distribution of n XX T, which will be a key ingredient of our later derivation for EEG n,n 2 p. The main approach employed to prove the assertion is called moment approach. It can be shown that for each k N, x k dφ XX x x k df n T y x 8 We refer the reader to the seminal survey by Bai [26] for further details on the moment approach and the Marčenko Pastur Law-like results. The following two lemmas will be needed. Lemma 2. In Page 29 [27], Let µ be a measure. Suppose that functions g n, h n, f n converge almost everywhere to functions g, h, f, respectively, and that g n f n h n almost everywhere. If g n dµ gdµ and hn dµ hdµ, then f n dµ fdµ. Lemma 3. Weyl s inequality [28] Let X R n n, Y R n n and Z R n n be symmetric matrices such that X Y + Z. Suppose their eigenvalues are ordered as λ X λ 2 X λ n X, λ Y λ 2 Y λ n Y, and λ Z λ 2 Z λ n Z, respectively. Then for any i n. λ i Y + λ n Z λ i X λ i Y + λ Z, Recall that the random bipartite graph G n,n 2 p consists of all bipartite graphs with vertex set V V V 2, in which the edges connecting vertices between V and V 2 are chosen independently with probability p,. We assume V i n i i, 2, n + n 2 n and n 2 /n : y, ]. For brevity, let A n AG n,n 2 p be the adjacency matrix, and denote by Q n q ij R n n a quasi-unit matrix, where q ij q ji if i, j V or i, j V 2, and q ij q ji otherwise. Let

6 Symmetry 25, 7 22 I n R n n be the unit matrix and J n R n n be the matrix whose all entries are equal to. By labeling the vertices appropriately, we obtain X T Ā n : A n pj n Q n R n n, 9 X where X x ij R n 2 n is a random matrix with all entries x ij being independent and identically distributed with mean zero and variance p p. For λ R, by basic matrix transforms, namely, taking the determinants of both sides of λi n λi n X T λi n X T λ, X λi n2 λi n2 λ XX T X λi n2 we have λ n detλ 2 I n2 XX T λ n 2 detλi n Ān, where detm is the determinant of matrix M. Consequently, λ n n n 2 λ 2 det I n2 XX T n n λ n 2 det n n λ n I n Ān. n Therefore, the eigenvalues of Ān/ n are symmetric: If λ, then λ is the eigenvalue of Ān/ n if and only if λ 2 is the eigenvalue of XX T /n. Since XX T is positive semi-definite, we know that Ā n / n has at least n n 2 zero eigenvalues and its spectrum can be arranged in a non-increasing order as λ λ n2 λ n+ }{{} n n 2 λ n2 λ n λ, assuming n n 2 when n is large enough. In what follows, we shall investigate EEG n,n 2 p and prove Theorem through a series of propositions. For convenience, we sometimes write EEM : n i eλ im for a real symmetric matrix M R n n. Thus, EEG n,n 2 p EEA n. Proposition. e x dφ XX x e x df n T y x Proof. Let φ XXT be the density of Φ XXT. By means of Lemma, we get φ XX n n n T as n tends to infinity. It follows from the bounded convergence theorem that converges to f y e x dφ XX x e x df n T y x By Lemma we know that there exists a large ω < such that all eigenvalues of n XX T do not exceed ω. Since the expansion e x k xk /k! converges uniformly on [, ω], we obtain from 8 that ω e x dφ XX x e x df n T y x e x df y x

7 Symmetry 25, 7 22 Combining with we derive Proposition 2. EE Ān e x dφ XXT x n e x dφ XX x n T e x df y x e x df y x e x df y x e x dφ n XX T x b n n 2 + n 2 a 2 coshx b a x 2 x dx 2 + o n πp py where a and b are given as in Lemma. Proof. Define Then we have e λ + + e λ n2 Analogous to the proof of we derive Ψx n 2 { }. n λ 2 i XX n T x i n2 e x dψx n 2, e x dφ XXT x. 2 n e x dφ XX x e x df n T y x For any x [,, we have e x e x. By Lemma 2 and Proposition we deduce that Accordingly, we have e x dφ XX x e x df n T y x It follows from 2 and 3 that e x dφ XX x e x df n T y x 3 e λ + + e λ n2 n 2 e x df y x b a e x 2πp pxy πp py b e x a b xx adx b x a 2 x 2 dx 4

8 Symmetry 25, Next, we calculate the sum of the exponentials of the smallest n 2 eigenvalues of Ān/ n. Similarly as in 2, we obtain e λ n e λ n n 2 e x dψx n 2 Noting that e x e x for x [,, we likewise have e x dφ XXT x. 5 n e x dφ XX x e x df n T y x 6 by employing Lemma 2 and Proposition. It then follows from 5 and 6 that e λ n e λ n n 2 e x df y x b πp py x a 2 b e x a x 2 dx 7 Finally, combining 4, 7 and the fact that λ n2 + λ n, we readily deduce the assertion of Proposition 2. Proposition 3. n n 2 + n 2 e n b + e n a C πp py + o EE Ā n n n 2 + n 2 e n a + e n b C πp py + o, where a, b are given as in Lemma, and C b a b x 2 a x 2 dx. Proof. Define Ψx n 2 { }. n n 2 λ i XX n T x i Then we have Ψx Ψx/ n. Therefore, the sum of the exponentials of the largest n 2 eigenvalues of Ān is e λān + + e λn 2 Ān n 2 e x d Ψx n 2 e n x dψx Analogous to the proof of 4 we obtain n 2 e n x dφ n XX T x. eλ Ān + + e λn 2 Ān b e n x b πp py a x a dx 2 x 2 [ Ce ] n a πp py, Ce n b 8 πp py

9 Symmetry 25, where C b b a a x 2 x dx. 2 eigenvalues of Ān satisfies Similarly, the sum of the exponentials of the smallest n2 eλ n +Ān + + e λnān b e n x b πp py a x a dx 2 x 2 [ Ce n b πp py, Ce ] n a 9 πp py Combining 8, 9 and the fact that λ n2 +Ān λ n Ān, we complete the proof of Proposition 3. Proposition 4. e pn e O n + O EEA n e pn e O n + O Proof. Since n n 2 for large n, by the Geršhgorin circle theorem we deduce pn λ n pj n Q n λ pj n Q n pn. In view of Lemma 3 and A n Ān + pj n Q n, we get λ i Ān + λ n pj n Q n λ i A n λ i Ān + λ pj n Q n for all i. Consequently, e pn EEĀn EEA n e pn EEĀn. It then follows from Proposition 3 that e pn n n 2 + e pn n 2 e n b + e n a C πp py + o EE A n e pn n n 2 + e pn n 2 e n a + e n b C πp py + o 2 Note that the rate of convergence in 8 as well as 9 can be bounded by On using the moment approach for instance, see Theorem in [29] and the estimates in [26] pp Hence, the infinitesimal quantity o on both sides of 2 is equivalent to On. Inserting n n/ + y and n 2 yn into 2, we have e pn e O n + O EEA n e pn e O n + O as desired. With Proposition 4 in our hands, we quickly get the proof of our main result. Proof of Theorem. Since n On, the upper bound in Proposition 4 yields EEG n,n 2 p e n p e O n + O On the other hand, the lower bound EEG n,n 2 p e n 2p e O n + o can be easily read out from Theorem 3. of [2].

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