Bounds on the <i>α</i>-Distance Energy and <i>α</i>-Distance Estrada Index of Graphs

Yang, Yang; Sun, Lizhu; Bu, Changjiang

doi:https://doi.org/10.1155/2020/9393521

Discrete Dynamics in Nature and Society

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 9393521 | https://doi.org/10.1155/2020/9393521

Bounds on the α-Distance Energy and α-Distance Estrada Index of Graphs

Yang Yang,¹Lizhu Sun,²and Changjiang Bu^1,2

Academic Editor: Manuel De la Sen

Received31 Oct 2019

Revised27 Jan 2020

Accepted03 Feb 2020

Published29 Apr 2020

Abstract

Let G be a simple undirected connected graph, then is called the α-distance matrix of G, where , is the distance matrix of G, and is the vertex transmission diagonal matrix of G. In this paper, we study some bounds on the α-distance energy and α-distance Estrada index of G. Furthermore, we establish the relation between α-distance Estrada index and α-distance energy.

1. Introduction

1.1. α-Distance Energy of Graphs

In this paper, we suppose that G is a connected graph. Let be a graph with the vertex set and edge set . The distance between two vertices is the length of the shortest path between and , denoted by . The Wiener index of the graph G is . The matrix is called the distance matrix of G, where , . For some properties of distance matrix, see [1–3].

The adjacency matrix of the graph G is , where if and otherwise. The Laplacian matrix and signless Laplacian matrix of G are , respectively, where and is the degree of , .

In 2013, the study of Laplacian matrix and signless Laplacian matrix was extended to distance Laplacian matrices and distance signless Laplacian matrices defined as in equation (1) (see [4]). In 2016, the study of the spectrum of signless Laplacian matrix was generalized to a convex combination of and defined as (see [5]). In [6], the above study was further extended to the α-distance matrices (see equation (2)).

Let is called the transmission of . Letwhere . and are called the distance Laplacian matrix and distance signless Laplacian matrix of the graph G, respectively. A graph G is said to be transmission regular if the transmissions of all the vertices in are equal (see [4]). For a transmission regular graph G, the characteristic polynomials of and were characterized in [4]. For more properties of and , see [7–9].

In [6], the α-distance matrix of a graph Gwas defined. Clearly, , , , and . The spectra of is called the α-distance spectra of G. Since is a real symmetric matrix, the eigenvalues of are real. Let be the eigenvalues of . And let denote the spectral radius of . From the Perron–Frobenius theorem, we have . The spectral properties of were recently studied including spectral radius, second largest eigenvalue, k-th smallest eigenvalue, and smallest eigenvalue (see [6, 10–12]).

Graph energy is an important graph invariant in graph theory; some graph energies , , and are called the energy (original energy), the distance energy, and the distance signless Laplacian energy, respectively, where , , and denote the eigenvalues of , , and , respectively, and (see [13–16]).

Graph energy has important applications in the fields of mathematics and chemistry. There are many research studies on the above kinds of graph energy. Scholars gave the bounds on the energy of graphs, for example, the McClelland’s bounds [17], Koolen–Moulton’s bounds [18] and so on [19]. In [16], the distance energy of some graphs was calculated.

In [11], Guo and Zhou extended the concept of graph energy to a more general form called α-distance energy:where is the eigenvalue of , , . Clearly, and .

1.2. α-Distance Estrada Index of Graphs

In [20], a spectral quantity is put forward by Estrada. is called the Estrada index of G, where denote the eigenvalues of (see [20]). It is well-known that the Estrada index plays an important role in the problem of characterizing the molecular structure [21] and complex networks [22–25]. In [26], the study was extended to distance matrices, and the distance Estrada index of G is , where are the eigenvalues of .

In this paper, we consider a more general Estrada index. Letbe the α-distance Estrada index of G, where are the eigenvalues of . Clearly, .

1.3. Main Work

In this paper, we study some bounds on the α-distance energy and α-distance Estrada index of graphs in terms of the parameter α and the vertex number, the transmission of vertices and Wiener index. Furthermore, we establish the relation between α-distance Estrada index and α-distance energy.

2. Some Bounds for the α-Distance Energy of Graphs

To begin with this section, we introduce some notations and propositions.

Proposition 1 (see [6]). Let G be a graph with n vertices. Then,where and denotes the eigenvalues of .

In the following, a new matrix is established:where denotes identity matrix of order n. Let denote the eigenvalues of . Obviously,

Proposition 2. Let G be a graph with n vertices. Then,where and denote the eigenvalues of .

Proof. In order to prove equation (8), let denote the eigenvalues of , by equations (5) and (6), we haveBy equation (5), we haveThen,In the following, we introduce some Lemmas which are helpful for the following proofs of theorems.

Lemma 1 (see [6]). Let G be a graph with n vertices. Then,the equality holds if and only if G is a transmission regular graph.

Lemma 2 (see [27]). Let G be a graph with n vertices. Then,the equality holds if and only if . denotes a complete graph with n vertices.

Next, we give some bounds for the α-distance energy of a graph by using the parameter α and the vertex number.

Theorem 1. Let G be a connected graph with n vertices. Then,the equality holds if and only if .

Proof. Let be the eigenvalues of .
By Lemma 1 and , we know that . Suppose that ι is the largest number such that . It follows from equation (5) thatFrom Lemmas 1 and 2, we haveThe above three inequalities are the equality holds if and only if .

We give some bounds for α-distance energy through the order n, the transmission of vertex and the parameter α based on Cauchy–Schwarz inequalities in the following.

Theorem 2. Let G be a graph with n vertices. Then,where and .

Proof. Let denote the eigenvalues of . From Cauchy–Schwarz inequality, we haveUsing equations (7) and (8), we haveSo,Similarly, from equation (11), we knowAccording to arithmetic-geometric inequality, we haveBy equations (7) and (9), we havewhere .
So,In the following, we can obtain another lower bound in terms of the vertex number and the maximum value of of .

Corollary 1. Let G be a graph with n vertices, thenwhere and .

Proof. Let be a nonincreasing sequence of . From Cauchy–Schwarz inequality, we haveBy equation (7), we haveSo,

In the following, we obtained some new bounds for α-distance energy through the Ozeki [28] and Polya’s [29] inequality, respectively.

Lemma 3 (see [29]). Suppose and are real numbers for , thenwhere , , , and .

Lemma 4 (see [28]). If and are real numbers for , thenwhere , , , and .

Theorem 3. Let G be a graph with n vertices. Then,where and are the largest and the smallest of , respectively, and .

Proof. Let be a nonincreasing sequence of and let and , where . By Lemma 3, we haveBy equation (8), we haveThus,

Theorem 4. Let G be a graph with n vertices. Then,where and are the largest and the smallest of , respectively, and .

Proof. Let be a nonincreasing sequence of . According to Lemma 4, let and , where , we haveBy equation (7), we haveThen,

Lemma 5 (see [30]). Let be n real numbers. Then,where .

It follows from the above Proposition the following result holds directly.

Proposition 3. For a graph G with n vertices, let be the largest eigenvalue of . For ,

3. Bounds for the α-Distance Estrada Index of Graphs

In this section, some bounds for α-distance Estrada index are obtained in terms of Wiener index, the transmission of the vertex, spectral radius of , and the vertex number. Furthermore, we give the relation between α-distance Estrada index and α-distance energy.

Next, we establish some bounds on the α-distance Estrada index.

Lemma 6 (see [31]). Let be nonnegative real numbers. Then, for ,

Theorem 5. Let G be a graph with n vertices. Then,where .

Proof. Let be the eigenvalues of . From equation (5) and Lemma 6, we havewhere .

In the following, we obtained a lower bound on the α-distance Estrada index by arithmetic-geometric inequality.

Theorem 6. Let G be a graph with n vertices. Then,

Proof. Let be the eigenvalues of . Then,From arithmetic-geometric inequality and equation (5), we obtainBy means of a power-series expansion, we have

By substituting equations (47) and (48) in equation (46), we see that

Theorem 7. Let G be a graph with n vertices. Then,

Proof. Let , where Obviously, is a decreasing function when , and is increasing when . Then, , that is,and the equality holds if and only if . So, by this function and equation (5), we havewhere are the eigenvalues of .
Let , where . Clearly, is an increasing function when . From Lemma 1, we haveThen,Hence,From Theorem 7, we have the following result.

Corollary 2. Let G be a transmission regular graph with n vertices. Let for each . Then,

We are inspired by literature [32], and we give Theorems 8 and 9 as follows.

Lemma 7 (see [33]). For and such that . Then,where . Equality holds if and only if .

Theorem 8. Let G be a graph with n vertices. Then,where . Equality holds if and only if .

Proof. Let , for , for . Obviously, , and according to Lemma 7, we havewhere .
By equation (5), we haveSincethenEquality holds, that is, if and only if .

Lemma 8 (see [34]). For . Then,where .

Theorem 9. Let G be a graph with n vertices. Then,

Proof. Let for , by Lemma 8, we havewhere .
Analyzing the left and right side of the previous inequality, respectively, by equation (5), we have

Also, the relation between α-distance Estrada index and α-distance energy are established.

Theorem 10. Let G be a graph with n vertices. Then,

Proof. By the definition of α-distance energy, we have

4. Conclusion

It is well-known that the graph energy and the Estrada index of graphs are important topics in graph theory. The α-distance matrix is an extension of the distance matrix. And the α-distance energy and α-distance Estrada index are generalized distance energy and distance Estrada index of graphs, respectively. In this paper, we establish some bounds on α-distance energy and α-distance Estrada index of G. Furthermore, a new lower bound for the α-distance Estrada index in relation to the α-distance energy of the graph G is given.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare to have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11801115 and 11601102), the Natural Science Foundation of the Heilongjiang Province (no. QC2018002), and the Fundamental Research Funds for the Central Universities.

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Copyright © 2020 Yang Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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