1. 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) [
1]. Starting with the Königsberg seven-bridge problem in 1736, 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
and an edge set
. The adjacency matrix of
G is a symmetric
-matrix
, where
if vertices
i and
j are adjacent, and
otherwise. It is well-known in algebraic graph theory that
has exactly
n real eigenvalues
due to its symmetry. They are usually called the spectrum (eigenvalues) of
G itself [
2].
A spectral graph invariant, the Estrada index
of
G, is defined as
This quantity was introduced by Estrada [
3] in 2000. It has noteworthy chemical applications, such as quantifying the degree of folding of long-chain molecules and the Shannon entropy [
3,
4,
5,
6,
7]. The Estrada index provides a remarkable measure of subgraph centrality as well as fault tolerance in the study of complex networks [
1,
8,
9,
10]. Building upon varied symmetric features in graphs, mathematical properties of this invariant can be found in, e.g., [
11,
12,
13,
14,
15,
16,
17,
18].
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
. In the past few years, researchers managed to establish a number of lower and upper bounds to estimate this invariant (see [
13] 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
for most graphs with respect to other graph parameters such as the number of vertices
n.
The classical Erdős–Rényi random graph model
includes the edges between all pairs of vertices independently at random with probability
p [
19]. It has symmetric, bell-shaped degree distribution, which is shared by many other random graph models. Regarding the Estrada index, Chen
et al. [
20] showed the following result: Let
be a random graph with a constant
, then
Here, we say that a certain property
holds in
almost surely (a.s.) if the probability that a random graph
has the property
converges to 1 as
n tends to infinity. Therefore, the result (
1) 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 [
21,
22,
23]. Formally, a bipartite graph is a graph whose vertices can be divided into two disjoint sets
and
such that every edge connects a vertex in
to a vertex in
. 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
, where
for
, satisfying
.
The authors in [
20] posed the following conjecture pertaining to bipartite graphs.
Conjecture 1. Let be a random bipartite graph with a constant . Thenif and only if . In this paper, by means of the symmetry in
and the spectral distribution of random matrix, we obtain lower and upper bounds for
. For ease of analysis, we assume that
. We establish the estimate
Thus a weak version of Conjecture 1 follows readily:
holds, provided
(
i.e.,
).
3. Proof of Theorem 1
This section is devoted to the proof of Theorem 1, which heavily relies on the symmetry in .
Throughout the paper, we shall understand
as a constant. Let
be a random matrix, where the entries
are independent and identically distributed with
. We denote by the eigenvalues of
M by
and its empirical spectral distribution by
Lemma 1. (Marčenko–Pastur Law [
26])
Let be a random matrix, where the entries are independent and identically distributed with mean zero and variance . Suppose that are functions of n, and . Then, with probability 1, the empirical spectral distribution converges weakly to the Marčenko–Pastur Law as , where has the densityand has a point mass at the origin if , where and . The above result formulates the limit spectral distribution of
, which will be a key ingredient of our later derivation for
. The main approach employed to prove the assertion is called moment approach. It can be shown that for each
,
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 219 [
27]),
Let μ be a measure. Suppose that functions converge almost everywhere to functions , respectively, and that almost everywhere. If and , then . Lemma 3. (Weyl’s inequality [
28])
Let and be symmetric matrices such that . Suppose their eigenvalues are ordered as , , and , respectively. Thenfor any . Recall that the random bipartite graph consists of all bipartite graphs with vertex set , in which the edges connecting vertices between and are chosen independently with probability . We assume (), and .
For brevity, let
be the adjacency matrix, and denote by
a quasi-unit matrix, where
if
or
, and
otherwise. Let
be the unit matrix and
be the matrix whose all entries are equal to 1. By labeling the vertices appropriately, we obtain
where
is a random matrix with all entries
being independent and identically distributed with mean zero and variance
. For
, by basic matrix transforms, namely, taking the determinants of both sides of
we have
where
is the determinant of matrix
M. Consequently,
Therefore, the eigenvalues of
are symmetric: If
, then
is the eigenvalue of
if and only if
is the eigenvalue of
. Since
is positive semi-definite, we know that
has at least
zero eigenvalues and its spectrum can be arranged in a non-increasing order as
assuming
when
n is large enough.
In what follows, we shall investigate
and prove Theorem 1 through a series of propositions. For convenience, we sometimes write
for a real symmetric matrix
. Thus,
.
Proof. Let
be the density of
. By means of Lemma 1, we get
converges to
a.s. as
n tends to infinity. It follows from the bounded convergence theorem that
By Lemma 1 we know that there exists a large
such that all eigenvalues of
do not exceed
ω. Since the expansion
converges uniformly on
, we obtain from (
8) that
Combining (
10) with (
11) we derive
□
Proposition 2. where a and b are given as in Lemma 1. Proof. Define
Then we have
Analogous to the proof of (
10) we derive
For any
, we have
. By Lemma 2 and Proposition 1 we deduce that
Accordingly, we have
It follows from (
12) and (
13) that
Next, we calculate the sum of the exponentials of the smallest
eigenvalues of
. Similarly as in (
12), we obtain
Noting that
for
, we likewise have
by employing Lemma 2 and Proposition 1. It then follows from (
15) and (
16) that
Finally, combining (
14), (
17) and the fact that
, we readily deduce the assertion of Proposition 2. □
Proposition 3. where a, b are given as in Lemma 1, and . Proof. Define
Then we have
. Therefore, the sum of the exponentials of the largest
eigenvalues of
is
Analogous to the proof of (
14) we obtain
where
. Similarly, the sum of the exponentials of the smallest
eigenvalues of
satisfies
Combining (
18), (
19) and the fact that
, we complete the proof of Proposition 3. □
Proof. Since
for large
n, by the Geršhgorin circle theorem we deduce
In view of Lemma 3 and
, we get
for all
i. Consequently,
. It then follows from Proposition 3 that
Note that the rate of convergence in (
18) as well as (
19) can be bounded by
using the moment approach (for instance, see Theorem 4.5.5 in [
29] ) and the estimates in [
26] (pp. 621–623). Hence, the infinitesimal quantity
on both sides of (
20) is equivalent to
. Inserting
and
into (
20), we have
as desired. □
With Proposition 4 in our hands, we quickly get the proof of our main result.
Proof of Theorem 1. Since
, the upper bound in Proposition 4 yields
a.s. On the other hand, the lower bound
≥
a.s. can be easily read out from Theorem 3.1 of [
20]. □
We mention that it is possible that our method can be pushed further to obtain sharper bounds, for example, a more fine-grained analysis in Proposition 2 could give better estimates for the second order terms in the expansion for .