Improve some of bounds for the Randić Estrada index of graphs

Let G be a graph with n vertices and let ρ1, ρ2, . . . , ρn be the eigenvalues of Randić matrix. The Randić Estrada index of G is REE(G) = ∑n i=1 e ρi . In this paper, we establish new lower and upper bounds for Randić index in terms of graph invariants such as the number of vertices and eigenvalues of graphs and improve some previously published lower bounds.

the spectral radius of the graph G. The graph invariant EE defined by was introduced by Estrada [7] in 2000, it is called Estrada index. Estrada index was first used to quantify the degree of folding of protein. It has also been used to measure centrality of complex networks. Applications of the Estrada index in quantum chemistry [8] and in information theory [5] can also be found in the literature. In addition to this, in [8] 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 ( [1], [6], [10], [11], [12], [13], [14], [20]).
For any graph G of order n, the Randić matrix of G, R = (r ij ), is defined in [2] as follows: if the vertices v i and v j of G are adjacent 0 otherwise.
The Randić eigenvalues ρ 1 , ρ 2 , . . . , ρ n of the graph G are the eigenvalues of its Randić matrix. Since R(G) is real symmetric matrix, its eigenvalues are real number. So we can order them so that ρ 1 ≥ ρ 2 ≥ · · · ≥ ρ n .
Then the Randić Estrada index of the graph G is defined in [3] as following: Recall that the general Randić index of a graph G is defined in [4] as following: The general Randić index when α = −1 is We make use of the following results in this paper. Lemma 1. [15] Let a 1 , a 2 , ..., a n ≥ 0 and p 1 , p 2 , ..., p n ≥ 0 such that .
Equality holds if and only if x = 0.
Let tr denotes the trace of a Randić matrix and ρ 1 , ρ 2 , ..., ρ n be the eigenvalues of the Randić matrix R(G), recall that Lemma 5. [17] Let G be a graph of order n and Randić matrix R. Then (4) Theorem 1. [19] Let A n (x) and G n (x) be the arithmetic and the geometric mean, respectively. Then

Motivation
Given that the Estrade index is one of the first and most applications topological indices. Recently, the Randić index has come to the attention of many researchers and also, we all know that the Randić index has wide applications in chemistry and physics. Maden in [17], obtained some of bounds for Randić Estrada index and Bozkurt et al. [3], discussed on the Randić Estrada index. In this paper, we decided to improve the published bounds of the Randić Estrada index.

Bounds for the Randić Estrada index
In this section, we establish some lower bounds for the Randić Estrada index in terms of graph invariants such as the order and eigenvalues. First we present a lower bound for Randić Estrada index that this bound improve bound in Theorem 1 for n ≥ 3.

Theorem 2.
Let G be a connected graph of order n. Then with equality if and only if x = 1. By the definition of Randić Estrada index and Inequality (5), we get: Lemma 3 we get the result.
If G is bipartite, the property of R-eigenvalues of G is similar to that of A-eigen values of G, hence ρ 1 = −ρ n . In the following result, we obtain a sharp lower bound of the Randić Estrada index for a bipartite graph.

Theorem 3. Let G be a connected bipartite graph of order n. Then
Proof. By the definition of Randić Estrada index and Inequality (5), we have Since, is an increasing function on D f = [0, +∞), hence we get the result.
The bound of Theorem 3 improves the next results for all connected bipartite graphs G with ∆(G) ≥ n 2 .

Theorem 4. [17] Let G be a connected bipartite graph with n vertices. Then
The next result presented in [17], is an immediate consequence of Theorem 2.
By Equalities (1)-(4), we have the following equations: Combining the above relations, we get as desired.
Next, we obtain a lower bound for the Randić Estrada index in terms of the order and the eigenvalues of its Randić matrix. + e ρ 1 − n + 1.

Now derive a lower bound and an upper bound of the Randić Estrada index in
terms of the order n and the eigenvalues of its Randić matrix.

Theorem 7.
Let G be a graph of order n. Then Equalities holds if and only if G is the empty graph K n .
Proof. Setting a i = e ρ i for i = 1, 2, ..., n and applying Lemma 2, we have Therefore by the definition of Randić Estrada index, we have Again by Lemma 2, we obtain By the definition of Randić Estrada index, we have The equality holds if and only if all ρ i are zero that is G is K n .
Theorem 8. Let G be a graph of order n. Then Furthermore, this bound is sharp for K n .
Proof. Setting b = e ρ 1 and a = e ρn , x i = e ρ i for i = 1, 2, ..., n and applying Lemma 1, we have and this leads to the desired bound.

Conclusions
In this paper, At first we obtain new bounds for Randić Estrada index and improved some of the important published bounds.