1 Introduction

Throughout this paper we consider simple graphs, that is, finite and undirected graphs without loops and multiple edges. If \(G\) is a graph with vertex set \(\{1, \ldots , n\}\), the adjacency matrix of \(G\) is an \(n\times n\) matrix \(A=[a_{ij}]\), where \(a_{ij} = 1\) if there is an edge between the vertices \(i\) and \(j\), and 0 otherwise. Since \(A\) is a real symmetric matrix, its eigenvalues \(\lambda _1,\lambda _2,\ldots ,\lambda _n\) are real numbers. These are referred to as the eigenvalues of \(G\). In what follows we assume that \(\lambda _1\ge \lambda _2\ge \cdots \ge \lambda _n\). The multiset of eigenvalues of \(A\) is called the spectrum of \(G\). For details of the theory of graph spectra see [2, 3]. We denote the complete graph on \(n\) vertices by \(K_n\), the complete bipartite graph whose parts are of orders \(a,b\) by \(K_{a,b}\).

The energy of \(G\) is defined as [13]

$$\begin{aligned} {\mathrm{E}}(G) = \sum _{i=1}^n|\lambda _i|. \end{aligned}$$

For details on graph energy see the reviews [14, 16, 18], the recent papers [17, 19] and the references cited therein.

The Estrada index of \(G\), recently put forward by Ernesto Estrada [6–8], is defined as

$$\begin{aligned} {\mathrm{EE}}(G) = \sum _{i=1}^n e^{\lambda _i}. \end{aligned}$$

Although invented in year 2000 [6], the Estrada index has already found a remarkable variety of applications. Initially it was used to quantify the degree of folding of long-chain molecules, especially proteins [6–8]; for this purpose the EE-values of pertinently constructed weighted graphs were employed. Another, fully unrelated, application of EE (of simple graphs) was proposed by Estrada and Rodríguez-Velázquez [10, 11]. They showed that EE provides a measure of the centrality of complex (communication, social, metabolic, etc.) networks. In addition to this, in [12] a connection between EE and the concept of extended atomic branching was considered. An application of the Estrada index in statistical thermodynamic has also been reported [9].

Mathematical properties of the Estrada index were studied in a number of recent works [4, 15]; for a review see [5].

In this paper we find a lower bound for the Estrada index of a graph in terms of the number of vertices, edges and triangles and two lower bounds terms of energy. These bounds improve previous bound given in [1, 4].

2 A Lower Bound in Terms of Number of Vertices, Edges and Triangles

In this section we give a lower bound for Estrada index of a graph in terms of the number of vertices, edges and triangles which is a significant improvement of the following bound.

Theorem 1

([4]) Let \(G\) be graph with \(n\) vertices, \(m\) edges and \(t\) triangles. Then

$$\begin{aligned} {\mathrm{EE}}(G)\ge \sqrt{n^2+4m+8t}. \end{aligned}$$

Equality holds if and only if \(G\) is the empty graph \(\overline{K}_n\).

Recall that ([2]) for a graph with eigenvalues \(\lambda _1,\lambda _2,\ldots ,\lambda _n\), with \(m\) edges and \(t\) triangles,

$$\begin{aligned} \sum _{i=1}^n\lambda _i&=0, \end{aligned}$$
(1)
$$\begin{aligned} \sum _{i=1}^n\lambda _i^2&=2m,\end{aligned}$$
(2)
$$\begin{aligned} \sum _{i=1}^n\lambda _i^3&=6t. \end{aligned}$$
(3)

Lemma 1

For any real \(x\), one has \(e^x\ge 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}.\) Equality holds if and only if \(x=0\).

Proof

By the Taylor theorem, for any \(x\ne 0\), there is a real \(\eta \ne 0\) between \(x\) and 0 such that \(e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{\eta ^4}{4!}\). This proves the lemma. \(\square \)

Theorem 2

Let \(G\) be graph with \(n\) vertices, \(m\) edges and \(t\) triangles. Then

$$\begin{aligned} {\mathrm{EE}}(G)\ge \sqrt{n^2+2mn+2nt}. \end{aligned}$$

Equality holds if and only if \(G\) is the empty graph \(\overline{K}_n\).

Proof

Suppose that \(\lambda _1,\lambda _2,\ldots ,\lambda _n\) is the spectrum of \(G\). Using Lemma 1 we have

$$\begin{aligned} {\mathrm{EE}}(G)^2&=\sum _{i=1}^n\sum _{j=1}^n e^{\lambda _i+\lambda _j}\\&\ge \sum _{i=1}^n\sum _{j=1}^n \left( 1+\lambda _i+\lambda _j+\frac{(\lambda _i+\lambda _j)^2}{2}+\frac{(\lambda _i+\lambda _j)^3}{6}\right) \\&=\sum _{i=1}^n\sum _{j=1}^n \left( 1+\lambda _i+\lambda _j+\lambda _i^2/2+\lambda _j^2/2+\lambda _i\lambda _j+\lambda _i^3/6+\lambda _j^3/6+\lambda _i^2 \lambda _j/2+\lambda _i\lambda _j^2/2\right) . \end{aligned}$$

Now, by (1),

$$\begin{aligned} \sum _{i=1}^n\sum _{j=1}^n (\lambda _i+\lambda _j)&=n\sum _{i=1}^n\lambda _i+n\sum _{j=1}^n \lambda _j=0,\\ \sum _{i=1}^n\sum _{j=1}^n \lambda _i\lambda _j&=(\sum _{i=1}^n\lambda _i)^2=0,\\ \sum _{i=1}^n\sum _{j=1}^n (\lambda _i^2\lambda _j/2+\lambda _i\lambda _j^2/2)&=\frac{1}{2}\sum _{i=1}^n\lambda _i^2\cdot \sum _{j=1}^n \lambda _j+\frac{1}{2}\sum _{i=1}^n\lambda _i\cdot \sum _{j=1}^n \lambda _j^2=0. \end{aligned}$$

By (2),

$$\begin{aligned} \sum _{i=1}^n\sum _{j=1}^n (\lambda _i^2/2+\lambda _j^2/2)=\frac{n}{2}\sum _{i=1}^n\lambda _i^2+\frac{n}{2}\sum _{j=1}^n \lambda _j^2=2mn. \end{aligned}$$

Similarly by (3),

$$\begin{aligned} \sum _{i=1}^n\sum _{j=1}^n (\lambda _i^3/6+\lambda _j^3/6)=2nt. \end{aligned}$$

Combining the above relations, we get

$$\begin{aligned} {\mathrm{EE}}(G)^2\ge n^2+2mn+2nt. \end{aligned}$$

So the inequality of the theorem is proved. By Lemma 1 equality holds if and only if all \(\lambda _i\) are zero that is \(G\) is \(\overline{K}_n\). \(\square \)

3 Lower Bounds in Terms of Energy

Recently, in [1] the following were proved.

Theorem 3

([1]) Let \(p\), \(\eta \) and \(q\) be, respectively, the number of positive, zero and negative adjacency eigenvalues of \(G\). Then

$$\begin{aligned} {\mathrm{EE}}(G)\ge \eta + pe^{{\mathrm{E}}(G)/(2p)} + qe^{-{\mathrm{E}}(G)/(2q)}. \end{aligned}$$

Equality holds if and only if \(G\) is either

  1. (i)

    a union of complete bipartite graphs \(K_{a_1,b_1}\cup \cdots \cup K_{a_p,b_p}\) with (possibly) some isolated vertices, such that \(a_1b_1=a_2b_2=\cdots =a_pb_p\), or

  2. (ii)

    a union of copies of \(K_{k\times t}\), for some fixed positive integers \(k,t\), with (possibly) some isolated vertices.

Theorem 4

([1]) If \(G\) is a bipartite graph, then \({\mathrm{EE}}(G)\ge \eta +r\cosh \left( \frac{{\mathrm{E}}(G)}{r}\right) \), where \(r\) is the rank of the adjacency matrix of \(G\). Equality holds if and only if \(G\) is a union of complete bipartite graphs \(K_{a_1,b_1}\cup \cdots \cup K_{a_p,b_p}\) with (possibly) some isolated vertices, such that \(a_1b_1=a_2b_2=\cdots =a_pb_p\).

We improve these lower bounds as follows.

Theorem 5

Let \(G\) be a graph with largest eigenvalue \(\lambda _1\) and let \(p\), \(\eta \) and \(q\) be, respectively, the number of positive, zero and negative eigenvalues of \(G\). Then

$$\begin{aligned} {\mathrm{EE}}(G)\ge e^{\lambda _1} + \eta + (p-1)e^\frac{{\mathrm{E}}(G)-2\lambda _1}{2(p-1)} + qe^{-\frac{{\mathrm{E}}(G)}{2q}}. \end{aligned}$$
(4)

Equality holds if and only if \(G\) is a graph such that all negative eigenvalues and all positive eigenvalues but the largest are equal, i.e. the spectrum of \(G\) is of the form \(\{[\lambda _1],[\theta _1]^{p-1},[0]^\eta ,[\theta _2]^q\}\), with \(\lambda _1\ge \theta _1>0>\theta _2\), where the exponents show the multiplicities.

Proof

Let \(\lambda _1\ge \cdots \ge \lambda _p\) be the positive, and \(\lambda _{n-q+1},\ldots ,\lambda _n\) be the negative eigenvalues of \(G\). As the sum of eigenvalues of a graph is zero, one has

$$\begin{aligned} {\mathrm{E}}(G)=2\sum _{i=1}^p \lambda _i=-2\sum _{i=n-q+1}^n \lambda _i. \end{aligned}$$

By the arithmetic–geometric mean inequality, we have

$$\begin{aligned} \sum _{i=2}^p e^{\lambda _i} \ge (p-1)e^{(\lambda _2+\cdots +\lambda _p)/(p-1)}=(p-1)e^\frac{{\mathrm{E}}(G)/2-\lambda _1}{p-1}. \end{aligned}$$
(5)

Similarly,

$$\begin{aligned} \sum _{i=n-q+1}^n e^{\lambda _i} \ge qe^{-{\mathrm{E}}(G)/(2q)}. \end{aligned}$$
(6)

For the zero eigenvalues, we also have

$$\begin{aligned} \sum _{i=p+1}^{n-q}e^{\lambda _i}=\eta . \end{aligned}$$

So we obtain

$$\begin{aligned} {\mathrm{EE}}(G)\ge e^{\lambda _1}+\eta +(p-1)e^\frac{{\mathrm{E}}(G)/2-\lambda _1}{p-1}+qe^{-\frac{{\mathrm{E}}(G)}{2q}}. \end{aligned}$$

The equality holds in (4) if and only if equality holds in both (5) and (6) and these happen if and only if \(\lambda _2=\cdots =\lambda _p\) and \(\lambda _{n-q+1}=\cdots =\lambda _n\). This completes the proof. \(\square \)

Theorem 5 can be improved for bipartite graphs to the following.

Theorem 6

If \(G\) is a bipartite graph, then

$$\begin{aligned} {\mathrm{EE}}(G)\ge \eta +2\cosh (\lambda _1)+(r-2)\cosh \left( \frac{{\mathrm{E}}(G)-2\lambda _1}{r-2}\right) , \end{aligned}$$
(7)

where \(r\) is the rank of the adjacency matrix of \(G\) and \(\lambda _1\) is the largest eigenvalue of \(G\). Equality holds if and only if the spectrum of \(G\) is of the form \(\{[\pm \lambda _1],[\pm \lambda _2]^{p-1},[0]^\eta \}\), with \(\lambda _1\ge \lambda _2>0\).

Proof

Since \(G\) is bipartite, its eigenvalues are symmetric with respect to zero, i.e. \(\lambda _i=-\lambda _{n-i+1}\) for \(i=1,\ldots ,\lfloor n/2\rfloor \). With a similar argument as the proof of Theorem 5, we find that

$$\begin{aligned} {\mathrm{EE}}(G)&=\eta +e^{\lambda _1}+e^{-\lambda _1}+\sum _{i=2}^p e^{\lambda _i}+\sum _{i=2}^p e^{-\lambda _i}\\&\ge \eta +e^{\lambda _1}+e^{-\lambda _1}+(p-1)e^\frac{{\mathrm{E}}(G)/2-\lambda _1}{p-1}+(p-1)e^{-\frac{{\mathrm{E}}(G) /2-\lambda _1}{p-1}}. \end{aligned}$$

Since the rank of adjacency matrix of \(G\) is equal to \(2p\), (7) follows.

Equality holds in (7) if and only if \(\lambda _2=\cdots =\lambda _p\). The proof is now complete. \(\square \)