Bounds on the $\alpha$-distance spectrum of graphs

TL;DR

This paper introduces bounds on the spectral radius, energy, and Estrada index related to the $ ext{alpha}$-distance matrix of graphs, expanding spectral graph theory with new spectral bounds and indices.

Contribution

It defines the $ ext{alpha}$-distance Estrada index and provides bounds on its spectral radius, energy, and Estrada index, advancing the understanding of spectral properties of $ ext{alpha}$-distance matrices.

Findings

Bounds on the spectral radius of $D_{\alpha}(G)$

Bounds on the $\alpha$-distance energy of $G$

Bounds on the $\alpha$-distance Estrada index

Abstract

For a simple, undirected and connected graph $G$, $D_{\alpha}(G) = \alpha Tr(G) + (1-\alpha) D(G)$ is called the $\alpha$-distance matrix of $G$, where $\alpha\in [0,1]$, $D(G)$ is the distance matrix of $G$, and $Tr(G)$ is the vertex transmission diagonal matrix of $G$. Recently, the $\alpha$-distance energy of $G$ was defined based on the spectra of $D_{\alpha}(G)$. In this paper, we define the $\alpha$-distance Estrada index of $G$ in terms of the eigenvalues of $D_{\alpha}(G)$. And we give some bounds on the spectral radius of $D_{\alpha}(G)$, $\alpha$-distance energy and $\alpha$-distance Estrada index of $G$.

Equations

$$\tilde{L}(G)=\tilde{D}(G)-\tilde{A}(G)\text{mbox}and\tilde{Q}(G)=\tilde{D}(G)+\tilde{A}(G),$$

$$Tr(v_i)=\sum _{v_j\in V(G),i\ne j}d(v_i,v_j).$$

$$\mathcal{L}(G)=Tr(G)-D(G)\text{mbox}and\mathcal{Q}(G)=Tr(G)+D(G),$$

$$D_{\alpha }(G)=\alpha Tr(G)+(1-\alpha )D(G),\alpha \in [0,1],$$

$${\varsigma }_{\alpha }(G)=\sum _{i=1}^n|{\sigma }_i(G)-\frac{2\alpha W(G)}{n}|,\alpha \in [0,1],$$

$$t(G)=\frac{1}{n}\sum _{i=1}^{n}Tr(v_i),$$

$${\rho }_{\alpha }(G)\ge \frac{2W(G)}{n},$$

$${\rho }_{\alpha }(T)⩾{\rho }_{\alpha }(S_n)=\frac{(\alpha +2)n-4+\sqrt{[(\alpha +2)n-4{]}^2+4[(n-1)(2\alpha -2n\alpha +1)]}}{2},$$

$$W(G)\ge \frac{n(n-1)}{2},$$

$${\varsigma }_{\alpha }(G)\ge 2(1-\alpha )(n-1),\alpha \in [1/2,1),$$

$$\sum _{i=1}^n{\sigma }_i(G)=\mathrm{Trace}(D_{\alpha }(G))=\sum _{i=1}^n\alpha Tr(v_i)=n\alpha t(G)=2\alpha W(G).$$

$${\varsigma }_{\alpha }(G)$$

$$=\sum _{i=1}^{\iota }{\sigma }_i(G)-\iota \alpha t(G)+(n-\iota )\alpha t(G)-\sum _{i=\iota +1}^n{\sigma }_i(G)$$

$$=\sum _{i=1}^{\iota }{\sigma }_i(G)-\iota \alpha t(G)+\sum _{i=1}^n{\sigma }_i(G)-\iota \alpha t(G)-\sum _{i=\iota +1}^n{\sigma }_i(G)$$

$$=2\sum _{i=1}^{\iota }({\sigma }_i(G)-\alpha t(G))$$

$$\ge 2({\sigma }_1(G)-\alpha t(G)).$$

$$2({\sigma }_1(G)-\alpha t(G))\ge 2(\frac{2W(G)}{n}-\alpha \frac{2W(G)}{n})=4(1-\alpha )\frac{W(G)}{n}\ge 2(1-\alpha )(n-1),$$

$${\varsigma }_{\alpha }(S_n)=$$

$$\frac{\alpha (-3n^2+8n-4)+2n^2-4n+n\sqrt{(\alpha -2{)}^2{n}^2+8\alpha n-12n-8\alpha +12}}{2n}$$

$$+|\frac{\alpha (-3n^2+8n-4)+2n^2-4n-n\sqrt{(\alpha -2{)}^2{n}^2+8\alpha n-12n-8\alpha +12}}{2n}|$$

$$+|3\alpha -2-\frac{2\alpha }{n}|(n-2).$$

$${\varsigma }_{\alpha }(S_n)$$

$$=\frac{\alpha (-3n^2+8n-4)+2n^2-4n+n\sqrt{(\alpha -2{)}^2{n}^2+8\alpha n-12n-8\alpha +12}}{2n}$$

$$+|\frac{\alpha (-3n^2+8n-4)+2n^2-4n-n\sqrt{(\alpha -2{)}^2{n}^2+8\alpha n-12n-8\alpha +12}}{2n}|$$

$$+|3\alpha -2-\frac{2\alpha }{n}|(n-2).$$

$$\Vert M{\Vert }_F=\sqrt{\sum _{i=1}^n\sum _{j=1}^n|m_{i,j}{|}^2}.$$

$$\sum _{i=1}^n{\xi }_i^2(G)=2|\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\xi }_i(G){\xi }_j(G)|=\Vert D(G){\Vert }_F^2,$$

$$\sum _{i=1}^n{\sigma }_i^2(G)=\Vert D_{\alpha }(G){\Vert }_F^2={\alpha }^2\sum _{i=1}^{n}T{r}^2(v_i)+(1-\alpha {)}^2\Vert D(G){\Vert }_F^2,$$

$$\Vert D(G){\Vert }_F^2<\frac{1}{n}(\sum _{i=1}^{n}Tr(v_i){)}^2.$$

$$\Vert D_{\alpha }(G){\Vert }_F<\sqrt{{\alpha }^2\sum _{i=1}^{n}T{r}^2(v_i)+(1-\alpha {)}^2\frac{1}{n}(\sum _{i=1}^{n}Tr(v_i){)}^2}.$$

$${\varsigma }_{\alpha }(G)\le \sqrt{nZ(G)},$$

$$\sum _{i=1}^n({\eta }_i{)}^2$$

$$=\sum _{i=1}^n({\sigma }_i(G)-\frac{2\alpha W(G)}{n}{)}^2$$

$$=\sum _{i=1}^n({\sigma }_i(G){)}^2-\sum _{i=1}^n\frac{4\alpha {\sigma }_i(G)W(G)}{n}+\sum _{i=1}^n\frac{4{\alpha }^2{W}^2(G)}{n^2}$$

$$=(1-\alpha {)}^2\Vert D(G){\Vert }_F^2+{\alpha }^2\sum _{i=1}^{n}T{r}^2(v_i)-\frac{4\alpha W(\alpha )}{n}\sum _{i=1}^n{\sigma }_i(G)+\sum _{i=1}^n\frac{4{\alpha }^2{W}^2(G)}{n^2}$$

$$=(1-\alpha {)}^2\Vert D(G){\Vert }_F^2+{\alpha }^2\sum _{i=1}^{n}T{r}^2(v_i)-\frac{4\alpha W(\alpha )}{n}\sum _{i=1}^n\alpha Tr(v_i)+\sum _{i=1}^n\frac{4{\alpha }^2{W}^2(G)}{n^2}$$

$$=(1-\alpha {)}^2\Vert D(G){\Vert }_F^2+{\alpha }^2\sum _{i=1}^n(Tr(v_i)-\frac{2W(G)}{n}{)}^2.$$

$$({\varsigma }_{\alpha }(G){)}^2=(\sum _{i=1}^n|{\eta }_i|{)}^2\le \sum _{i=1}^n({\eta }_i(G){)}^2\sum _{i=1}^{n}1=nZ(G).$$

Full text

1 Introduction

1.1 Distance spectrum of graphs

In this paper, we consider the simple, undirected and connected graphs. Let $G=(V(G)$, $E(G))$ be a graph with the vertex set $V(G)=\{v_{1},\ldots,v_{n}\}$ and edge set $E(G)$. The distance between two vertices $v_{i},v_{j}\in V(G)$ is the length of the shortest path between $v_{i}$ and $v_{j}$, denoted by $d(v_{i},v_{j})$. The Wiener index $W(G)$ of the graph $G$ is the sum of the distances between all pairs of vertices in $G$, i.e., $W(G)=\frac{1}{2}\sum_{v_{i},v_{j}\in V(G),i\neq j}d(v_{i},v_{j}).$ The matrix $D(G)=(d_{i,j})\in\mathbb{R}^{n\times n}$ with entries $d_{i,j}=d(v_{i},v_{j})$ is called the distance matrix of $G$, where $i,j\in\{1,2,\cdots,n\}$. The spectra of $D(G)$ is called the distance spectra of $G$.

The study of the distance spectra arises from the work of R. Graham and H.O. Pollack in 1971 (see [11]). In [11], the determinant of the distance matrix of a tree $T$ was given as $\det(D(T))=(-1)^{n-1}(n-1)2^{n-2}$, where $n=|V(T)|$. And Graham and Pollack proved $D(T)$ has $1$ positive eigenvalue and $n-1$ negative eigenvalues. In 1978, Graham and Lovász established the representation for the inverse of $D(T)$ (see [10]). The distance spectrum of graphs has attracted much attention [27].

1.2 $\alpha$-spectrum of graphs

The adjacency matrix of the graph $G$ is $\widetilde{A}(G)=(\widetilde{a}_{ij})\in\mathbb{R}^{n\times n}$, where $\widetilde{a}_{ij}=1$ if $(i,j)\in E(G)$, and $\widetilde{a}_{ij}=0$ otherwise. The Laplacian matrix and signless Laplacian matrix of $G$ are

[TABLE]

respectively, where $\widetilde{D}(G)=\mathrm{diag}(\widetilde{d}_{v_{1}},\cdots,\widetilde{d}_{v_{n}})\in\mathbb{R}^{n\times n}$ and $\widetilde{d}_{v_{i}}$ is the degree of $v_{i}$, $i=1,2,\cdots,n$.

It is well-known that the spectrum of the adjacency matrix, Laplacian matrix and signless Laplacian matrix of a graph were widely investigated [5]. In 2013, the study of the spectrum of Laplacian matrix and signless Laplacian matrix was extended to distance Laplacian matrices and distance signless Laplacian matrices defined as in Equation (1) (see [1]). In 2016, the study of the spectrum of Laplacian matrix and signless Laplacian matrix was generalized to a convex combination of $\widetilde{D}(G)$ and $\widetilde{A}(G)$ defined as $A_{\alpha}{(G)}=\alpha\widetilde{D}(G)+(1-\alpha)\widetilde{A}(G),~{}\alpha\in[0,1]$ (see [24]). Recently, the above study was further extended to the $\alpha$-distance matrices defined as in Equation (2) (see [3]).

For a vertex $v_{i}\in V(G)$, the sum of the distances between $v_{i}$ and all the other vertices in $V(G)$ is called the transimission of $v_{i}$, denoted by $Tr(v_{i})$, that is

[TABLE]

A graph $G$ is said to be transmission regular if the transimissions of all the vertices in $V(G)$ are equal. In 2013, Aouchiche and Hansen [1] defined the distance Laplacian matrix $\mathcal{L}(G)$ and distance signless Laplacian matrix $\mathcal{Q}(G)$ of the graph $G$,

[TABLE]

where $Tr(G)=\mathrm{diag}(Tr(v_{1}),\cdots,Tr(v_{n}))$.

For a transmission regular graph $G$, the characteristic polynomials of $\mathcal{L}(G)$ and $\mathcal{Q}(G)$ were calculated in [1]. In [21,28,29], the bounds on the spectral radii for the distance signless Laplacian matrix of trees, unicyclic graphs and bicyclic graphs were characterized, respectively.

In [3], the $\alpha$-distance matrix of a graph $G$

[TABLE]

was defined. Clearly, $D_{0}(G)=D(G)$, $2D_{\frac{1}{2}}(G)=\mathcal{Q}(G)$ and $D_{1}(G)=Tr(G)$. The spectra of $D_{\alpha}(G)$ is called the $\alpha$-distance spectra of $G$. Since $D_{\alpha}(G)$ is a real symmetric matrix, the eigenvalues of $D_{\alpha}(G)$ are real. Let $\sigma_{1}(G)\geq\sigma_{2}(G)\geq\cdots\geq\sigma_{n}(G)$ be the eigenvalues of $D_{\alpha}(G)$. And let $\rho_{\alpha}(G)=\max\{\sigma_{i}(G)|i=1,2,\cdots,n\}$ be the spectral radius of $D_{\alpha}(G)$. We know that $D_{\alpha}(G)$ is a nonnegative weakly irreducible matrix. From the Perron-Frobenius Theorem, we have $\sigma_{1}(G)=\rho_{\alpha}(G)$. And $D_{\alpha}(G)$ is positive semidefiniteness for $\alpha\in[\frac{1}{2},1]$.

In [3], upper and lower bounds for the spectral radius of the $\alpha$-distance matrix were established. In [4], authors characterized the unique graph with minimum spectral radius of the $\alpha$-distance matrix among the connected graphs with fixed chromatic number. In [20], a lower bound on the $k$-th smallest eigenvalue of the $\alpha$-distance matrix was given. In [14], the bounds on the spectral radius of the $\alpha$-distance matrix were established.

1.3 $\alpha$-distance energy of graphs

Let $\lambda_{i}(G)$, $\kappa_{i}(G)$ and $\nu_{i}(G)$ be the eigenvalues of $\widetilde{A}(G)$, $D(G)$ and $\mathcal{Q}(G)$ respectively, where $i=1,2\cdots,n$ and $n=|V(G)|$. The energy of $G$ is $E_{\pi}(G)=\sum_{i=1}^{n}|\lambda_{i}(G)|$ (see [12,13]). The sum $DE(G)=\sum_{i=1}^{n}|\kappa_{i}(G)|$ is called the distance energy of $G$ (see [17]), and $DSLE(G)=\sum_{i=1}^{n}|\nu_{i}(G)-\frac{2W(G)}{n}|$ is called the distance signless Laplacian energy [6].

Graph energy has important applications in the fields of mathematics and chemistry. There are many researches on the above kinds of graph energy. Scholars gave the bounds on the energy of graphs, for example the McClelland${}^{\prime}s$ bounds [23], Koolen-Moulton${}^{\prime}s$ bounds [18] and so on [2]. In [17], the distance energy of some graphs were calculated.

Recently, Zhou extended the concept of graph energy to a more general form called $\alpha$-distance energy

[TABLE]

where $\sigma_{i}(G)$ is the eigenvalue of $D_{\alpha}(G)$, $i=1,2,\cdots,n$, $n=|V(G)|$ (see [14]). Clearly, $\varsigma_{0}(G)=DE(G)$ and $\varsigma_{1/2}(G)=\frac{1}{2}DSLE(G)$.

1.4 Main work

In this paper, we give some bounds on the $\alpha$-distance energy of graphs and stars in terms of the parameter $\alpha$ and the vertex number. And we establish some bounds for the spectral radius of the $\alpha$-distance matrix by using the transimission of vertices and Wiener index. Further, we define the $\alpha$-distance Estrada index of a graph, and obtain some bounds on the $\alpha$-distance Estrada index.

2 Some bounds for the $\alpha$-distance energy of graphs

To begin with this section, we introduce some notations. The average transmission of the graph $G$, denoted by $t(G)$, is defined by

[TABLE]

where $n=|V(G)|$. Clearly, $t(G)=\frac{2W(G)}{n}.$ Let $S=\sum_{1\leq i<j\leq n}d^{2}(v_{i},v_{j})$ (see [3]).

Lemma 2.1**.**

[3]* Let $G$ be a graph with $n$ vertices. Then*

[TABLE]

the equality holds if and only if $G$ is a transmission regular graph.

For a graph $G$ and $v\in V(G)$, let $N_{G}(v)=|\{u:(v,u)\in E(G)\}|$.

Lemma 2.2**.**

[3]* Let $G$ be a graph with $n$ vertices. Let $S$ be a subset of $V(G)$ such that $N_{G}(x)=N_{G}(y)$ for any $x,y\in S$.

(1) If $S$ is an independent set, then $Tr(v)$ is a constant $h$ for each $v\in S$, and $D_{\alpha}(G)$ has $\alpha(h+2)-2$ as an eigenvalue with multiplicity at least $|S|-1$.

(2) If $S$ is a clique, then $Tr(v)$ is a constant $h^{\ast}$ for each $v\in S$, and $D_{\alpha}(G)$ has $\alpha(h^{\ast}+1)-1$ as an eigenvalue with multiplicity at least $|S|-1$.*

From Lemma 2.2, we have the following result.

Proposition 2.3**.**

For a graph $G$, $D_{\alpha}(G)$ ($\alpha\in[0,1)$) has two distinct eigenvalues if and only if $G$ is a complete graph.

Let $S_{n}$ be a star with $n$ vertices.

Lemma 2.4**.**

[4,14,20]* Let $T$ be a tree with $n\geqslant 4$ vertices. Then*

[TABLE]

the equality holds if and only if $G\cong S_{n}$.

Proposition 2.5**.**

*The $\alpha$-distance spectra of $S_{n}$ consists of

(1) $(2n-1)\alpha-2$ with multiplicity $n-2$;

(2) $\frac{\alpha n+2n-4\pm\sqrt{(\alpha-2)^{2}n^{2}+8\alpha n-12n-8\alpha+12}}{2}$.*

Proof.

Since $S_{n}$ satisfies the statement (1) of Lemma 2.2, and the number of the vertices in the independent set is $n-1$ and the transimission of each vertex in the independent set is $2n-3$, we have $(2n-1)\alpha-2$ is an eigenvalue of $D_{\alpha}(S_{n})$ with multiplicity at least $n-2$. From Lemma 2.4, the statement (2) holds. ∎

Let $K_{n}$ be a complete graph with $n$ vertices.

Lemma 2.6**.**

[9]* Let $G$ be a graph with $n$ vertices. Then*

[TABLE]

the equality holds if and only if $G\cong K_{n}$.

Next, we give some bounds for the $\alpha$-distance energy of a graph and characterize the $\alpha$-distance energy of $S_{n}$ by using the parameter $\alpha$ and the vertex number.

Theorem 2.7**.**

Let $G$ be a connected graph with $n$ vertices. Then

[TABLE]

the equality holds if and only if $G\cong K_{n}$.

Proof.

Let $\sigma_{1}(G)\geq\sigma_{2}(G)\geq\cdots\geq\sigma_{n}(G)$ be the eigenvalues of $D_{\alpha}(G)$. It is easy to see that

[TABLE]

Lemma 2.1 gives $\sigma_{1}(G)\geq t(G)\geq\alpha t(G)$. Suppose that $\iota$ is the largest number such that $\sigma_{\iota}(G)\geq\alpha t(G)$. And it follows from Equation (3) that

[TABLE]

From Lemmas 2.1 and 2.6, we have

[TABLE]

and $\varsigma_{\alpha}(G)=2(1-\alpha)(n-1)$ if and only if $G\cong K_{n}$. ∎

Theorem 2.8**.**

The $\alpha$-distance energy of $S_{n}$ is

[TABLE]

Proof.

We know that $t(S_{n})=2n-4+\frac{2}{n}.$ And from Proposition 2.5, we obtain

[TABLE]

∎

Let $\|M\|_{F}$ denote the Frobenius norm of a matrix $M=(m_{i,j})\in\mathbb{C}^{n\times n}$, that is

[TABLE]

We know that $\|M\|_{F}^{2}=\sum_{i=1}^{n}|\tilde{\lambda}_{i}(M)|^{2}$, where $\tilde{\lambda}_{1}(M),\ldots,\tilde{\lambda}_{n}(M)$ are the eigenvalues of $M$. For a graph $G$ with $n$ vertices,

[TABLE]

where $\xi_{1}(G),\xi_{2}(G),\ldots,\xi_{n}(G)$ are the eigenvalues of $D(G)$ (see [7,8]). Similarly to the above equation, we have

[TABLE]

where $\sigma_{1}(G),\sigma_{2}(G),\ldots,\sigma_{n}(G)$ are the eigenvalues of $D_{\alpha}(G)$.

Lemma 2.9**.**

[8]* Let $G$ be a graph with $n$ vertices. Then*

[TABLE]

From Lemma 2.9, the following result can be obtained directly.

Proposition 2.10**.**

Let $G$ be a connected graph with $n$ vertices. Then

[TABLE]

In the following, we establish some bounds for the $\alpha$-distance energy of a graph $G$ by using the Frobenius norm of $D(G)$, transmissions of vertices and $W(G)$.

Theorem 2.11**.**

Let $G$ be a graph with $n$ vertices. Then

[TABLE]

where $Z(G)=(1-\alpha)^{2}\|D(G)\|_{F}^{2}+\alpha^{2}\sum_{i=1}^{n}(Tr(v_{i})-\frac{2W(G)}{n})^{2}$.

Proof.

Let $\eta_{i}(G)=\sigma_{i}(G)-\frac{2\alpha W(G)}{n}$, where $i=1,\ldots,n$. Clearly, $\sum_{i=1}^{n}\eta_{i}(G)=0.$ From Equation (4), we have

[TABLE]

Let $Z(G)=(1-\alpha)^{2}\|D(G)\|_{F}^{2}+\alpha^{2}\sum_{i=1}^{n}(Tr(v_{i})-\frac{2W(G)}{n})^{2}.$ By Cauchy-Schwar$z^{\prime}$s inequality, we have

[TABLE]

∎

Lemma 2.12**.**

[3]* Let $G$ be a graph with $n$ vertices. Then*

[TABLE]

where $\sigma_{1}(G),\sigma_{2}(G),\ldots,\sigma_{n}(G)$ are the eigenvalues of $D_{\alpha}(G)$.

Theorem 2.13**.**

Let $G$ be a graph with $n$ vertices. Then

[TABLE]

Proof.

From Cauchy-Schwar$z^{\prime}$s inequality we have

[TABLE]

It follows from Lemma 2.12 that

[TABLE]

∎

3 Bounds for the $\alpha$-distance spectrum of graphs

Lemma 3.14**.**

[8]* Let $x_{1}>x_{2}\geq\ldots\geq x_{n}>0$ be $n$ real numbers. Then*

[TABLE]

where $M=\frac{\sum_{1=1}^{n}x_{i}}{n}$.

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

Proposition 3.15**.**

For a graph $G$ with $n$ vertices, let $\sigma_{1}(G)$ be the largest eigenvalue of $D_{\alpha}(G)$. For $\alpha\in[\frac{1}{2},1)$,

[TABLE]

For a matrix $M=(m_{ij})\in\mathbb{C}^{n\times n}$, let $R_{i}(M)=\sum_{j=1}^{n}m_{ij}$ be the $i$-th row sum of $M$, $i=1,2,\cdots,n$.

Lemma 3.16**.**

[7,19]* Let $A=(a_{i,j})\in\mathbb{R}^{n\times n}$ be a nonnegative matrix. Let $\rho(A)$ be the spectral radius of $A$. Then*

[TABLE]

Further, if $A$ is an irreducible matrix, then the above two equalities hold if and only if $R_{1}(A)=R_{2}(A)=\cdots=R_{n}(A)$.

From the above Lemma, it is easy to see the following result holds.

Proposition 3.17**.**

Let $G$ be a graph with $n$ vertices. Let $\rho(D_{\alpha}(G))$ be the spectral radius of $D_{\alpha}(G)$. Then

[TABLE]

the equality holds if and only if $G$ is a transmission regular graph.

Lemma 3.18**.**

[22]* For a graph $G$ with $n$ vertices, let $\lambda_{1}(G)$ be the largest eigenvalue of the signless Laplacian matrix $\widetilde{Q}(G)$. Let $p(x)$ be a polynomial on $x$. Then*

[TABLE]

Moreover, if the row sums of $p(\tilde{Q}(G))$ are not all equal, then both inequalities are strict.

Inspired by the above result, we give the following bounds on the largest eigenvalue of $D_{\alpha}(G)$.

Proposition 3.19**.**

For a graph $G$ with $n$ vertices, let $\sigma_{1}(G)$ be the largest eigenvalue of $D_{\alpha}(G)$. Let $p(x)$ be a polynomial on $x$. For $\alpha\in[\frac{1}{2},1)$,

[TABLE]

If the row sums of $p(D_{\alpha}(G))$ are not all equal, then both inequalities are strict.

Proof.

Perron-Frobenius Theorem gives that there exists a positive vector $\mathbf{x}=(x_{1},\ldots,x_{n})^{\rm T}$ such that $D_{\alpha}(G)\mathbf{x}=\sigma_{1}(G)\mathbf{x}$. Then

[TABLE]

Let $\sum_{i=1}^{n}x_{i}=1$. Then

[TABLE]

Hence,

[TABLE]

∎

Lemma 3.20**.**

For a graph $G$ with $n$ vertices, let $\mathbf{T}=\max\{Tr(v)|{v\in V(G)}\}$ and $\mathbf{t}=\min\{Tr(v)$ $|{v\in V(G)}\}$. Then for each vertex $u\in V(G)$,

[TABLE]

Proof.

By calculation, we have

[TABLE]

Similarly,

[TABLE]

∎

Theorem 3.21**.**

For a graph $G$ with $n$ vertices, let $\mathbf{T}=\max\{Tr(v)|{v\in V(G)}\}$ and $\mathbf{t}=\min\{Tr(v)$ $|{v\in V(G)}\}$. Let $\sigma_{1}(G)$ be the largest eigenvalue of $D_{\alpha}(G)$, where $\alpha\in[\frac{1}{2},1)$. Then for $u\in V(G)$,

[TABLE]

Proof.

Let $\mathbf{e}_{i}\in\mathbb{R}^{n}$ be an unity vector whose $i$-component is $1$. And let $\mathbf{1}\in\mathbb{R}^{n}$ be a vector whose components are all $1$. Then

[TABLE]

From Lemma 3.7, we have

[TABLE]

and

[TABLE]

Let $p(x)=x^{2}-(1-\alpha)(\mathbf{t}-1)x$. Then the sum of the entries in the $i$-th row of $p(D_{\alpha}(G))$ is

[TABLE]

Clearly,

[TABLE]

By Proposition 3.6, we obtain

[TABLE]

that is

[TABLE]

Similarly, let $p(x)=x^{2}-(1-\alpha)(\mathbf{T}-1)x$. Then

[TABLE]

∎

Lemma 3.22**.**

[26]* let $x_{1}\geq x_{2}\geq,\ldots,\geq x_{m}$ be real numbers such that $\sum_{i=1}^{m}x_{i}=0$. Then*

[TABLE]

the equality holds if and only if $x_{1}=\cdots=x_{m}=-\frac{x_{1}}{m-1}$.

Theorem 3.23**.**

Let $G$ be a graph with $n$ vertices. Then

[TABLE]

the equality holds if and only if $G\cong K_{n}$.

Proof.

Obviously,

[TABLE]

It follows from Lemma 3.9 that

[TABLE]

and the equality holds if and only if

[TABLE]

By calculation, we know

[TABLE]

Then

[TABLE]

and the equality holds if and only if $\sigma_{2}(G)-\frac{2\alpha W(G)}{n}=\cdots=\sigma_{n}(G)-\frac{2\alpha W(G)}{n}=-\frac{\sigma_{1}(G)-\frac{2\alpha W(G)}{n}}{n-1}$. And it follows from Proposition 2.3 that the equality holds if and only if $G\cong K_{n}$. ∎

For an $n\times n$ matrix $M$ and order partition $(K_{1},K_{2},\ldots,K_{m})$ of the ordered set $\{1,2,\ldots,n\}$, $M$ can be denoted as a partition matrix

[TABLE]

where $M_{ij}$ has $K_{i}$ as the set of its row indices and $K_{j}$ as the set of its column indices, Let $B_{M}$ be the quotient matrix of the partitioned matrix $M$, which is defined to be an $m\times m$ matrix with the $ij$-entry $(B_{M})_{ij}=\frac{1}{|K_{i}|}\sum_{l=1}^{K_{i}}\sum_{k=1}^{K_{j}}{(M_{ij})_{lk}}$, where $1\leq i,j\leq m$ (see [5] ).

Lemma 3.24**.**

[16]* Suppose $B_{M}\in\mathbb{C}^{m\times m}$ is a quotient matrix of a symmetric partitioned matrix $M\in\mathbb{C}^{n\times n}$. Let $\{\mu_{1},\cdots,\mu_{m}\}$ and $\{\lambda_{1},\cdots,\lambda_{n}\}$ be the eigenvalue sets of $B_{M}$ and $M$, respectively. Then for $i=1,2,\cdots,m$,*

[TABLE]

Proposition 3.25**.**

Let $G$ be a graph with $n$ vertices. Let $\sigma_{1}(G)$ and $\sigma_{n}(G)$ be the largest and the smallest eigenvalue of $D_{\alpha}(G)$, respectively. Then for each $v_{i}\in V(G)$,

[TABLE]

Proof.

Let $K_{1}=\{i\}$ and $K_{2}=\{V(G)\setminus i\}$ be a partition of $V(G)$. Then the following $B_{M}$ is the corresponding quotient matrix of $D_{\alpha}(G)$,

[TABLE]

Let $\{\sigma_{1}(G),\cdots,\sigma_{n}(G)\}$ and $\{\lambda_{1}(B_{M}),\lambda_{1}(B_{M})\}$ be the eigenvalue sets of $D_{\alpha}(G)$ and $B_{M}$, respectively. It follows from Lemma 3.11 that

[TABLE]

where $i=1,2$. Hence,

$\sigma_{1}(G)\geq\lambda_{1}(B_{M})$ and $\lambda_{2}(B_{M})\geq\sigma_{n}(G)$.

Then

[TABLE]

Since $\det(xI-M_{B})=0$, we obtain

[TABLE]

[TABLE]

[TABLE]

∎

Let $G$ be a graph with $n$ vertices. Let $\{\lambda_{1}(G),\lambda_{2}(G),\cdots,\lambda_{n}(G)\}$ be the eigenvalue set of $\widetilde{A}(G)$. And $EE(G)=\sum_{i=1}^{n}e^{\lambda_{i}(G)}=\sum_{i=1}^{n}\sum_{k=0}^{\infty}\frac{\lambda_{i}(G)^{k}}{k!}$ is called the Estrada index of $G$. It is well-known that Estrada index plays an important role in the problem of characterizing the molecular structure [30]. In [15], the study was extended to distance matrices, and the distance Estrada index of $G$ was defined by $DEE(G)=\sum_{i=1}^{n}e^{o_{i}(G)}=\sum_{i=1}^{n}\sum_{k=0}^{\infty}\frac{o_{i}(G)^{k}}{k!}$, where $\{o_{1}(G),o_{2}(G),\ldots,o_{n}(G)\}$ is the eigenvalue set of $D(G)$.

In this paper, we consider a more general Estrada index. Let

[TABLE]

be the $\alpha$-distance Estrada index of $G$, where $\{\sigma_{1}(G),\cdots,\sigma_{n}(G)\}$ is the eigenvalue set of $D_{\alpha}(G)$. Clearly, $DEE_{0}(G)=DEE(G)$. Next, we establish some bounds on the $\alpha$-distance Estrada index.

Lemma 3.26**.**

[25]* Let $x_{1},x_{2},\ldots,x_{n}$ be nonnegative real numbers. Then for $k\geq 2$,*

[TABLE]

Theorem 3.27**.**

Let $G$ be a graph with $n$ vertices. Then

[TABLE]

where $\omega=\sqrt{\alpha^{2}\sum_{i=1}^{n}Tr(v_{i})^{2}+2(1-\alpha)^{2}S}$.

Proof.

Let $\sigma_{1}(G),\ldots,\sigma_{n}(G)$ be the eigenvalues of $D_{\alpha}(G)$. From Lemmas 2.12 and 3.13, we have

[TABLE]

[TABLE]

where $\omega=\sqrt{\alpha^{2}\sum_{i=1}^{n}Tr(v_{i})^{2}+2(1-\alpha)^{2}S}$. ∎

Theorem 3.28**.**

Let $G$ be a graph with $n$ vertices. Then

[TABLE]

Proof.

Let $\sigma_{1}(G),\ldots,\sigma_{n}(G)$ be the eigenvalues of $D_{\alpha}(G)$. Then

[TABLE]

From Arithmetic-Geometric inequality, we obtain

[TABLE]

By Taylor expansion theorem, we have

[TABLE]

Let $\delta\in[0,4]$. Then

[TABLE]

Hence,

[TABLE]

It is elementary to show for $n\geq 2$, let the function

[TABLE]

where $0\leq\alpha\leq 1$, then $f(x)$ is monotonically decreasing in the interval $[0,4]$. Hence, $f(0)$ is a largest lower bound of $DEE_{\alpha}(G)$. ∎

Theorem 3.29**.**

Let $G$ be a graph with $n$ vertices. Then

[TABLE]

Proof.

Let $f(x)=(x-1)-\ln x$, where $x>0.$ Obviously, $f(x)$ is a decreasing function when $x\in(0,1]$, and $f(x)$ is increasing when $x\in[1,+\infty)$. Then $f(x)\geq f(1)=0$, that is

[TABLE]

and the equality holds if and only if $x=1$. So by this function, we have

[TABLE]

where $\sigma_{1}(G)>\sigma_{2}(G),\ldots,\geq\sigma_{k}(G)$ are the eigenvalues of $D_{\alpha}(G)$.

Let $\Gamma(x)=e^{x}+(n-1)+2\alpha W(G)-x$, where $x>0$. Clearly, $\Gamma(x)$ is an increasing function when $x\in(0,+\infty)$. From Lemma LABEL:lem1, we have

[TABLE]

then,

[TABLE]

Hence,

[TABLE]

∎

From Theorem 2.1, we have the following result.

Corollary 3.30**.**

Let $G$ be a transmission regular graph with $n$ vertices. Let $Tr(u)=r$ for each $u\in V(G)$. Then

[TABLE]

Next, a new upper bound for the $\alpha$-distance Estrada index is established.

Theorem 3.31**.**

Let $G$ be a graph with $n$ vertices. Then

[TABLE]

Proof.

By the definition of $\alpha$-distance energy, we have

[TABLE]

∎

4 Acknowledgments

Supported by by the National Natural Science Foundation of China (No. 11801115 and No.11601102), the Natural Science Foundation of the Heilongjiang Province (No.QC2018002) and the Fundamental Research Funds for the Central Universities.

References