Estimating the PI-Estrada index of graphs

Abstract Let G be a graph with n vertices. The PI-Estrada index of G is an invariant that is calculated from the eigenvalues of the vertex-PI matrix of G. The main purpose of this paper is to establish upper and lower bounds for the PI-Estrada index of a graph in terms of the number of vertices, edges, triangles and pendant vertices.


Introduction
Let G = (V, E) be a simple undirected graph with vertex set V (G) and edge set E(G). The integers n = |V | and m = |E| are the number of vertices and edges of the graph G, respectively. Let U be a subset of V (G). We denote by U the subgraph of G induced by U . A clique of G is a subset of mutually adjacent vertices of V (G). A clique is called maximal if it is not contained in any other clique. A clique is said to be maximum if it has the maximum cardinality. The size of a maximum clique in G is called the clique number of G and is denoted by ω(G). A walk from a vertex u to a vertex v is a finite alternating sequence v 0 (= u)e 1 v 1 e 2 . . . v k−1 e k v k (= v) of vertices and edges such that e i = v i−1 v i for i = 1, 2, . . . , k where the number k is the length of the walk. A graph is connected if each pair of vertices in a graph is joined by at least one walk. As usual, denote by P n , K n and K n the path, the complete graph, and the complement of complete graph with n vertices, respectively. The distance between two arbitrary vertices u and v of G, denoted by d (u, v), is defined as the number of edges in the shortest path connecting the vertices u and v. The adjacency matrix A(G) of G is a matrix with entries a ij = 1 if v i v j ∈ E(G) and 0 otherwise. We denote by λ 1 , λ 2 , . . . , λ n the eigenvalues of A(G).
The Estrada index, put forward by Ernesto Estrada [11], is defined as The Estrada index is used to quantify the degree of folding of long-chain molecules, especially proteins [11] and complex networks [29][30][31]. Additional applications of the Estrada index can be found in [7,[12][13][14][15]. The mathematical aspects of the Estrada index have been intensively studied. Several upper and lower bounds and new Estrada indices were obtained in [1,2,[25][26][27][28]32]. One of the most important properties of the Estrada index is as following: where M k = M k (G) is the k-th moment of the graph G, i.e., It is well known that M k (G) is equal to the number of closed walks of length k in G.
The investigation of topological indices has been shown to give a high degree of predictability of pharmaceutical properties. Among the several existing graph-based molecular structure descriptors [33,34], the Randić index is certainly the most widely applied in chemistry and pharmacology, in particular for designing in the quantitative structure-activity relationship models (QSPR) used in the chemical, biological science, and quantitative structure-property relationships (QSAR). Due to the importance and practicality of these topological indices, Bozkurt et al. [6] in 2012, introduced the Randić energy and the Randić Estrada index of a graph, which has been of interest to many researchers.
Let e = uv be an edge of G. Then, n u (e|G) is the number of edges lying closer to u than to v and n v (e|G) is defined analogously. The Padmakar-Ivan (PI) index of a graph G is defined [21] as For details on this invariant, see some of the most cited papers [19][20][21][22]24]. The PI index promises to be a useful parameter in the QSPR/QSAR studies. A more favorable comparison with other representative indices such as the Randić index has already been made in order to establish the predictive ability of the PI index, and the results have shown that in several cases the PI index gave better results. Considering the importance of this index and being better than the Randić index, Arani [3] in 2011, introduced the PI-energy. In this paper, we introduce the PI-Estrada index and examine some of its properties.
The vertex-PI matrix (see [3]) of G, denoted by P I(G), is defined as a matrix whose Since P I(G) is a real and symmetric matrix, its eigenvalues are real numbers, and we label them in non-increasing order by δ 1 δ 2 . . . δ n . Their collection is called the PI-spectrum of G. The spectral radius of G, denoted by δ 1 (G), is the largest eigenvalue of P I(G).
A spectral invariant of the matrix P I(G) is the vertex-PI energy of G, which is defined [3] as Recently, several analogous concepts such as Zagreb Estrada index [26], Harmonic Estrada index [18], Albertson-Estrada index [16] and Hermitian Estrada index [17] of graphs and digraphs were put forward.
On the other hand, the vertex-PI index is a distance-based molecular structure descriptor, that recently found numerous chemical applications. In this paper, we introduce the PI-Estrada index and denote it by EE P I . Definition 1.1. Let G be a graph of order n whose PI-eigenvalues are δ 1 δ 2 . . . δ n . The PI-Estrada index of G is defined as Also, we define Similar to the Estrada index in (1), we have This paper is organized as follows. In Section 2, we give a list of some previously known results. In Section 3, we establish upper and lower bounds for the PI-Estrada index.

Preliminaries and known results
In this section, we give some preliminary results useful for the proof of the our results. Arani et al. [4], proved the following result.
Lemma 2.1. [4] Let G be a graph with n vertices. Then Deng [9], showed the following result.

Lemma 2.4. [23]
Let T be a tree with n vertices, n 2. Then Lemma 2.5. [23] Let C n be a cycle graph with n 3. Then Equality holds if and only if x = 0.

Lemma 2.7.
For any non-negative numbers x 1 , x 2 , . . . , x n and k 2, we have Remark 2.1. For any real x, the power-series expansion of e x , is the following The following results come from [4].

Main results
In this section, we present upper and lower bounds for the PI-Estrada index in terms of the number of vertices, edges, triangles, a bound in terms of the elements of the degree sequence, pendent vertices and clique number for a graph, connected graphs, cycle graphs and tree graphs. Our first main result is the following.

Theorem 3.1. Let G be a graph with n vertices, m edges and t triangles. Then
Equality holds if and only if G is the empty graph K n .
Proof. Suppose that δ 1 , δ 2 , . . . , δ n form the spectrum of the P I matrix. By definition of the P I-Estrada index and Lemma 2.6, we have Now, by Equality (5), By Inequality (6), Also, by Equality (5) and Inequality (6), it holds that Similarly by inequality (7), Combining the above relations, we get EE P I (G) n 2 + 8n(t + m).
So, the inequality of the theorem is proved. The equality holds if and only if all δ i are zero that is G ∼ = K n .

Theorem 3.2. Let G be a graph with n vertices, m edges and t triangles. Then
where N 4 = n i=1 δ 4 i . Equality in (9) holds if and only if G is the empty graph K n .
8m. Therefore, by definition of the P I-Estrada index, we have The result follows easily. The equality holds if and only if all δ i are zero that is G ∼ = K n . Remark 3.1. Note that the above result can be written as: Equality holds if and only if G ∼ = K n .
Proof. Note that n i=1 δ 2 i 2mn 2 . By definition of the P I-Estrada index, we have Considering k 0 = 2 in (12), we have the following result that only depends on number of vertices and edges of the graph. If G ∼ = K n it is easy to check that the equality in (10) holds. Suppose now that the equality holds in (10). Then all the inequalities in the proof must be equalities. From Equation (3), we know that is valid if k even or all eigenvalues are negative, but we know that k cannot be just even, so all eigenvalues must be negative, by equality (5), we have n i=1 δ i = 0. Therefore, we get δ 1 = δ 2 = . . . = δ n = 0, hence G ∼ = K n . Now, by using Theorem 3.3, we get the next result.
Equality holds if and only if G ∼ = K 2 .
Proof. Let a i , b i are decreasing non-negative sequences with a 1 , b 1 = 0 and w i a nonnegative sequence, for i = 1, 2, . . . , n.
The following inequality is valid (see [10], p. 85) For a i = b i := |δ i |, and w i := 1, i = 1, 2, . . . , n, Inequality (14) becomes Then Therefore, by Lemma 2.9 and Inequality (6), we have It is not difficult to see that equality holds in (13) Theorem 3.4. Let G be a graph with n vertices and m edges. Then Equality holds if and only if G ∼ = K 2 .
Proof. By definition of the PI-Estrada index and using arithmetic-geometric mean inequality, we obtain EE P I (G) = e δ1 + e δ2 + · · · + e δn e δ1 + (n − 1) It is straightforward verified that f is an increasing function for x > 0. From the Lemma 3.1, we obtain It is not difficult to see that equality holds in (15) Analogous, Theorem 3.4 also by Lemmas 2.10 and 2.11, we have the next results.
Proof. By definition of the P I-Estrada index and Inequalities (3) and (6), we have

Concluding remarks
In this paper, considering the eigenvalues of the P I-matrix, upper and lower bounds are established for the P I-Estrada index of a graph as a function of various combinatorial and spectral invariants of the given graph. Considering this matrix, a future problem is to obtain a generalization for directed graphs.