# Bounds on theα-Distance Energy and α-Distance Estrada Index of Graphs

Authors: Yang Yang; Lizhu Sun; Changjiang Bu
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

LetG be a simple undirected connected graph, then DαG=αTrG+1−αDG is called the α-distance matrix of G, where α∈0,1, DG is the distance matrix of G, and TrG 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.


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## Body

## 1. Introduction

### 1.1.α-Distance Energy of Graphs

In this paper, we suppose thatG is a connected graph. Let G=VG,EG be a graph with the vertex set VG=v1,…,vn and edge set EG. The distance between two vertices vi,vj∈VG is the length of the shortest path between vi and vj, denoted by dvi,vj. The Wiener index WG of the graph G is WG=1/2∑vi,vj∈VGdvi,vj. The matrix DG=di,j∈Rn×n is called the distance matrix of G, where di,j=dvi,vj, i,j∈1,2,…,n. For some properties of distance matrix, see [1–3].Theadjacency matrix of the graph G is A˜G=a˜ij∈Rn×n, where a˜ij=1 if i,j∈EG and a˜ij=0 otherwise. The Laplacian matrix and signless Laplacian matrix of G are L˜G=D˜G−A˜GandQ˜G=D˜G+A˜G, respectively, where D˜G=diagd˜v1,…,d˜vn∈Rn×n and d˜vi is the degree of vi, i=1,2,…,n.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 D˜G and A˜G defined as AαG=αD˜G+1−αA˜G,α∈0,1 (see [5]). In [6], the above study was further extended to the α-distance matrices (see equation (2)).LetTrvi=∑vj∈VGdvi,vj is called the transmission of vi. Let(1)LG=TrG−DG,QG=TrG+DG,where TrG=diagTrv1,…,Trvn. LG and Q 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 VG are equal (see [4]). For a transmission regular graph G, the characteristic polynomials of LG and QG were characterized in [4]. For more properties of LG and QG, see [7–9].In [6], the α-distance matrix of a graph G(2)DαG=αTrG+1−αDG,α∈0,1,was defined. Clearly, D0G=DG, D1/2G=1/2QG, D1G=TrG, and DαG−DβG=α−βLG. The spectra of DαG is called the α-distance spectra of G. Since DαG is a real symmetric matrix, the eigenvalues of DαG are real. Let σ1αG≥σ2αG≥⋯≥σnαG be the eigenvalues of DαG. And let ραG denote the spectral radius of DαG. From the Perron–Frobenius theorem, we have σ1G=ραG. The spectral properties of DαG 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 energiesEπG=∑i=1nλiG, DEG=∑i=1nγiG, and DSLEG=∑i=1nνiG−2WG/n are called the energy (original energy), the distance energy, and the distance signless Laplacian energy, respectively, where λiG, γiG, and νiG denote the eigenvalues of A˜G, DG, and QG, respectively, i=1,2,…,n and n=VG (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:(3)ςαG=∑i=1nσiαG−2αWGn,α∈0,1,where σiαG is the eigenvalue of DαG, i=1,2,…,n, n=VG. Clearly, ς0G=DEG and ς1/2G=1/2DSLEG.


### 1.2.α-Distance Estrada Index of Graphs

In [20], a spectral quantity is put forward by Estrada. EEG=∑i=1neλiG=∑i=1n∑k=0∞λikG/k! is called the Estrada index of G, where λ1G,λ2G,…,λnG denote the eigenvalues of A˜G (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 DEEG=∑i=1neγiG=∑i=1n∑k=0∞γikG/k!, where γ1G,γ2G,…,γnG are the eigenvalues of DG.In this paper, we consider a more general Estrada index. Let(4)DEEαG=∑i=1neσiαG=∑i=1n∑k=0∞σiαGkk!,be the α-distance Estrada index of G, where σ1αG,…,σnαG are the eigenvalues of DαG. Clearly, DEE0G=DEEG.


### 1.3. Main Work

In this paper, we study some bounds on theα-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.


## 1.1.α-Distance Energy of Graphs

In this paper, we suppose thatG is a connected graph. Let G=VG,EG be a graph with the vertex set VG=v1,…,vn and edge set EG. The distance between two vertices vi,vj∈VG is the length of the shortest path between vi and vj, denoted by dvi,vj. The Wiener index WG of the graph G is WG=1/2∑vi,vj∈VGdvi,vj. The matrix DG=di,j∈Rn×n is called the distance matrix of G, where di,j=dvi,vj, i,j∈1,2,…,n. For some properties of distance matrix, see [1–3].Theadjacency matrix of the graph G is A˜G=a˜ij∈Rn×n, where a˜ij=1 if i,j∈EG and a˜ij=0 otherwise. The Laplacian matrix and signless Laplacian matrix of G are L˜G=D˜G−A˜GandQ˜G=D˜G+A˜G, respectively, where D˜G=diagd˜v1,…,d˜vn∈Rn×n and d˜vi is the degree of vi, i=1,2,…,n.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 D˜G and A˜G defined as AαG=αD˜G+1−αA˜G,α∈0,1 (see [5]). In [6], the above study was further extended to the α-distance matrices (see equation (2)).LetTrvi=∑vj∈VGdvi,vj is called the transmission of vi. Let(1)LG=TrG−DG,QG=TrG+DG,where TrG=diagTrv1,…,Trvn. LG and Q 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 VG are equal (see [4]). For a transmission regular graph G, the characteristic polynomials of LG and QG were characterized in [4]. For more properties of LG and QG, see [7–9].In [6], the α-distance matrix of a graph G(2)DαG=αTrG+1−αDG,α∈0,1,was defined. Clearly, D0G=DG, D1/2G=1/2QG, D1G=TrG, and DαG−DβG=α−βLG. The spectra of DαG is called the α-distance spectra of G. Since DαG is a real symmetric matrix, the eigenvalues of DαG are real. Let σ1αG≥σ2αG≥⋯≥σnαG be the eigenvalues of DαG. And let ραG denote the spectral radius of DαG. From the Perron–Frobenius theorem, we have σ1G=ραG. The spectral properties of DαG 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 energiesEπG=∑i=1nλiG, DEG=∑i=1nγiG, and DSLEG=∑i=1nνiG−2WG/n are called the energy (original energy), the distance energy, and the distance signless Laplacian energy, respectively, where λiG, γiG, and νiG denote the eigenvalues of A˜G, DG, and QG, respectively, i=1,2,…,n and n=VG (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:(3)ςαG=∑i=1nσiαG−2αWGn,α∈0,1,where σiαG is the eigenvalue of DαG, i=1,2,…,n, n=VG. Clearly, ς0G=DEG and ς1/2G=1/2DSLEG.


## 1.2.α-Distance Estrada Index of Graphs

In [20], a spectral quantity is put forward by Estrada. EEG=∑i=1neλiG=∑i=1n∑k=0∞λikG/k! is called the Estrada index of G, where λ1G,λ2G,…,λnG denote the eigenvalues of A˜G (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 DEEG=∑i=1neγiG=∑i=1n∑k=0∞γikG/k!, where γ1G,γ2G,…,γnG are the eigenvalues of DG.In this paper, we consider a more general Estrada index. Let(4)DEEαG=∑i=1neσiαG=∑i=1n∑k=0∞σiαGkk!,be the α-distance Estrada index of G, where σ1αG,…,σnαG are the eigenvalues of DαG. Clearly, DEE0G=DEEG.


## 1.3. Main Work

In this paper, we study some bounds on theα-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 theα-Distance Energy of Graphs

To begin with this section, we introduce some notations and propositions.Proposition 1 (see [6]).
LetG be a graph with n vertices. Then,(5)∑i=1nσiαG=∑i=1nαTrvi=2αWG,∑i=1nσiαG2=α2∑i=1nTr2vi+21−α2S,where S=∑1≤i<j≤nd2vi,vj and σ1αG≥σ2αG≥⋯≥σnαG denotes the eigenvalues of DαG.In the following, a new matrix is established:(6)UαG=αTrG+1−αDG−2αWGnIn,α∈0,1,where In denotes identity matrix of order n. Let η1αG,η2αG,…,ηnαG denote the eigenvalues of UαG. Obviously,(7)ςαG=∑i=1nηiαG,α∈0,1.Proposition 2.
LetG be a graph with n vertices. Then,(8)∑i=1nηiαG=0,∑i=1nηiαG2=2Z,(9)Z=−∑1≤i<j≤nηiαGηjαG=∑1≤i<j≤nηiαGηjαG,where Z=1−α2S+α2/2∑i=1nTrvi−2WG/n2 and η1αG,η2αG,…,ηnαG denote the eigenvalues of UαG.Proof.
In order to prove equation (8), let η1αG,η2αG,…,ηnαG denote the eigenvalues of UαG, by equations (5) and (6), we have(10)∑i=1nηiαG2=traceUα2G=21−α2S+α2∑i=1nTrvi−2WGn2.
By equation (5), we have(11)0=∑i=1nηiαG2=2Z+2∑1≤i<j≤nηiαGηjαG.
Then,(12)Z=−∑1≤i<j≤nηiαGηjαG=∑1≤i<j≤nηiαGηjαG.
In the following, we introduce some Lemmas which are helpful for the following proofs of theorems.Lemma 1 (see [6]).
LetG be a graph with n vertices. Then,(13)ραG≥2WGn,the equality holds if and only if G is a transmission regular graph.Lemma 2 (see [27]).
LetG be a graph with n vertices. Then,(14)WG≥nn−12,the equality holds if and only if G≅Kn. Kn denotes a complete graph with n vertices.Next, we give some bounds for theα-distance energy of a graph by using the parameter α and the vertex number.Theorem 1.
LetG be a connected graph with n vertices. Then,(15)ςαG≥21−αn−1,the equality holds if and only if G≅Kn.Proof.
Letσ1αG≥σ2αG≥…≥σnαG be the eigenvalues of DαG.
By Lemma1 and α∈0,1, we know that σ1αG≥2WG/n≥2WGα/n. Suppose that ι is the largest number such that σιαG≥2WGα/n. It follows from equation (5) that(16)ςαG=∑i=1ισiαG−2WGαn+∑i=ι+1n2WGαn−σiαG=∑i=1ισiαG−ι2WGαn+n−ι2WGαn−∑i=ι+1nσiαG=∑i=1ισiαG−ι2WGαn−ι2WGαn+∑i=1nσiαG−∑i=ι+1nσiαG=2∑i=1ισiαG−2WGαn≥2σ1αG−2WGαn.
From Lemmas1 and 2, we have(17)2σ1αG−2WGαn≥22WGn−α2WGn=41−αWGn≥21−αn−1.
The above three inequalities are the equality holds if and only ifG≅Kn.We give some bounds forα-distance energy through the order n, the transmission of vertex and the parameter α based on Cauchy–Schwarz inequalities in the following.Theorem 2.
LetG be a graph with n vertices. Then,(18)2Z+nn−1p2/n≤ςαG≤2nZ,where p=detUαG and Z=1−α2S+α2/2∑i=1nTrvi−2WG/n2.Proof.
Letη1αG,η2αG,…,ηnαG denote the eigenvalues of UαG. From Cauchy–Schwarz inequality, we have(19)ςαG2=∑i=1nηiαG⋅12≤∑i=1nηiαG2∑i=1n1.
Using equations (7) and (8), we have(20)ςαG2≤2nZ.
So,(21)ςαG≤2nZ.
Similarly, from equation (11), we know(22)ςαG2=∑i=1nηiαG2=∑i=1nηiαG2+∑i≠jηiαGηjαG.
According to arithmetic-geometric inequality, we have(23)1nn−1∑i≠jηiαGηjαG≥∏i≠jηiαGηjαG1/nn−1=∏i=1nηiαGn−1∏j=1nηjαGn−11/nn−1=∏i=1nηiαG2n−11/nn−1=∏i=1nηiαG2/n.
By equations (7) and (9), we have(24)ςαG2=∑i=1nηiαG2+∑i≠jηiαGηjαG≥2Z+nn−1p2/n,where p=detUαG.
So,(25)2Z+nn−1p2/n≤ςαG≤2nZ.
In the following, we can obtain another lower bound in terms of the vertex number and the maximum value ofηiαG of UαG.Corollary 1.
LetG be a graph with n vertices, then(26)ςαG≤δ1αG+n−12Z−δ1αG2,where δ1αG=maxηiαG and Z=1−α2S+α2/2∑i=1nTrvi−2WG/n2.Proof.
Letδ1αG≥δ2αG≥,…,≥δnαG be a nonincreasing sequence of ηiαG. From Cauchy–Schwarz inequality, we have(27)∑i=2nδiαG⋅12≤∑i=2nδiαG2∑i=2n1.
By equation (7), we have(28)ςαG−δ1αG2≤n−12Z−δ1αG2.
So,(29)ςαG≤δ1αG+n−12Z−δ1αG2.In the following, we obtained some new bounds forα-distance energy through the Ozeki [28] and Polya’s [29] inequality, respectively.Lemma 3 (see [29]).
Supposeai and bi are real numbers for 1≤i≤n, then(30)∑i=1nai2∑i=1nbi2≤14M1M2m1m2+m1m2M1M22∑i=1naibi2,where M1=max1≤i≤nai, M2=max1≤i≤nbi, m1=min1≤i≤nai, and m2=min1≤i≤nbi.Lemma 4 (see [28]).
Ifai and bi are real numbers for 1≤i≤n, then(31)∑i=1nai2∑i=1nbi2−∑i=1naibi2≤n24M1M2−m1m22,where M1=max1≤i≤nai, M2=max1≤i≤nbi, m1=min1≤i≤nai, and m2=min1≤i≤nbi.Theorem 3.
LetG be a graph with n vertices. Then,(32)ςαG≥22nZδ1αGδnαGδ1αG+δnαG,where δ1αG and δnαG are the largest and the smallest of ηiαG, respectively, and Z=1−α2S+α2/2∑i=1nTrvi−2WG/n2.Proof.
Letδ1αG≥δ2αG≥,…,≥δnαG be a nonincreasing sequence of ηiαG and let ai=δiαG and bi=1, where i=1,…,n. By Lemma 3, we have(33)∑i=1nδiαG2∑i=1n1≤14δ1αGδnαG+δnαGδ1αG2∑i=1nδiαG2.
By equation (8), we have(34)2nZ≤14δ1αGδnαG+δnαGδ1αG2ςαG2.
Thus,(35)ςαG≥22nZδ1αGδnαGδ1αG+δnαG.Theorem 4.
LetG be a graph with n vertices. Then,(36)ςαG≥2nZ−n24δ1αG−δnαG2,where δ1αG and δnαG are the largest and the smallest of ηiαG, respectively, and Z=1−α2S+α2/2∑i=1nTrvi−2WG/n2.Proof.
Letδ1αG≥δ2αG≥,…,≥δnαG be a nonincreasing sequence of ηiαG. According to Lemma 4, let ai=δiαG and bi=1, where i=1,…,n, we have(37)∑i=1nδiαG2∑i=1n1−∑i=1nδiαG⋅12≤n24δ1αG−δnαG2.
By equation (7), we have(38)2nZ−ςαG2≤n24δ1αG−δnαG2.
Then,(39)ςαG≥2nZ−n24δ1αG−δnαG2.Lemma 5 (see [30]).
Letx1>x2≥…≥xn>0 be n real numbers. Then,(40)∑i=1nxi−M<n2x1,where M=∑1=1nxi/n.It follows from the above Proposition the following result holds directly.Proposition 3.
For a graphG with n vertices, let η1αG be the largest eigenvalue of DαG. For α∈1/2,1,(41)ςαG≤n2η1αG.


## 3. Bounds for theα-Distance Estrada Index of Graphs

In this section, some bounds forα-distance Estrada index are obtained in terms of Wiener index, the transmission of the vertex, spectral radius of DαG, and the vertex number. Furthermore, we give the relation between α-distance Estrada index and α-distance energy.Next, we establish some bounds on theα-distance Estrada index.Lemma 6 (see [31]).
Letx1,x2,…,xn be nonnegative real numbers. Then, for k≥2,(42)∑i=1nxik≤∑i=1nxi2k/2.Theorem 5.
LetG be a graph with n vertices. Then,(43)DEEαG≤n+2αWG−1−ω+eω,where ω=α2∑i=1nTr2vi+21−α2S.Proof.
Letσ1αG,…,σnαG be the eigenvalues of DαG. From equation (5) and Lemma 6, we have(44)DEEαG=n+2αWG+∑i=1n∑k=2∞σiαGkk!≤n+2αWG+∑i=1n∑k=2∞σiαGkk!=n+2αWG+∑k=2∞1k!∑i=1nσiαGk≤n+2αWG+∑k=2∞1k!∑i=1nσiαG2k/2=n+2αWG+∑k=2∞1k!α2∑i=1nTr2vi+21−α2Sk/2=n+2αWG−1−ω+∑k=0∞1k!ωk=n+2αWG−1−ω+eω,where ω=α2∑i=1nTr2vi+21−α2S.In the following, we obtained a lower bound on theα-distance Estrada index by arithmetic-geometric inequality.Theorem 6.
LetG be a graph with n vertices. Then,(45)DEEαG≥n+4αWG+nn−1e4αWG/n.Proof.
Letσ1αG,…,σnαG be the eigenvalues of DαG. Then,(46)DEEαG2=∑i=1ne2σiαG+2∑1≤i<j≤neσiαGeσjαG.
From arithmetic-geometric inequality and equation (5), we obtain(47)2∑1≤i<j≤neσiαGeσjαG≥nn−1∏1≤i<j≤neσiαGeσjαG2/nn−1=nn−1∏i=1neσiαGn−12/nn−1=nn−1e∑i=1nσiαG2/n=nn−1e4αWG/n.
By means of a power-series expansion, we have(48)∑i=1ne2σiαG=∑i=1n∑k=0∞2σiαGkk!=n+4αWG+∑i=1n∑k=2∞2σiαGkk!≥n+4αWG.By substituting equations (47) and (48) in equation (46), we see that(49)DEEαG≥n+4αWG+nn−1e4αWG/n.Theorem 7.
LetG be a graph with n vertices. Then,(50)DEEαG≥e2WG/n+n−1+2αWG−2WGn.Proof.
Letfx=x−1−lnx, where x>0. Obviously, fx is a decreasing function when x∈0,1, and fx is increasing when x∈1,+∞. Then, fx≥f1=0, that is,(51)x≥1+lnx,x>0,and the equality holds if and only if x=1. So, by this function and equation (5), we have(52)DEEαG≥eσ1αG+n−1+∑k=2nlneσkαG=eσ1αG+n−1+∑k=2nσkαG=eσ1αG+n−1+2αWG−σ1αG,where σ1αG,…,σnαG are the eigenvalues of DαG.
LetΓx=ex+n−1+2αWG−x, where x>0. Clearly, Γx is an increasing function when x∈0,+∞. From Lemma 1, we have(53)σ1αG≥2WGn≥0.
Then,(54)Γσ1αG≥Γ2WGn.
Hence,(55)DEEαG≥e2WG/n+n−1+2αWG−2WGn.
From Theorem7, we have the following result.Corollary 2.
LetG be a transmission regular graph with n vertices. Let Tru=r for each u∈VG. Then,(56)DEEαG≥er+n−1+αnr−r.We are inspired by literature [32], and we give Theorems 8 and 9 as follows.Lemma 7 (see [33]).
Fora1,a2,…,an≥0 and p1,p2,…,pn≥0 such that ∑i=1npi=1. Then,(57)∑i=1npiai−∏i=1naipi≥nT1n∑i=1nai−∏i=1nai1/n,where T=minp1,p2,…,pn. Equality holds if and only if a1=a2=,…,=an.Theorem 8.
LetG be a graph with n vertices. Then,(58)DEEαG≥eσ1αG+2n−1Δ−n−1e2αWG/n,where Δ=e2n−1αWG+nσ2αG+,…,+σnαG/2nn−1. Equality holds if and only if G≅nK1.Proof.
Letp1=1/2n, pi=2n−1/2nn−1 for i=2,…,n, ai=eσiαG for i=2,…,n. Obviously, T=min1/2n,2n−1/2nn−1=1/2n, and according to Lemma 7, we have(59)eσ1αG2n+2n−12nn−1∑i=2neσiαG−Δ≥121n∑i=1neσiαG−∏i=1neσiαG/n,where Δ=eσ1αG/2n∏i=2ne2n−1σiαG/2nn−1.
By equation (5), we have(60)Δ=eσ1αG/2n∏i=2ne2n−1σiαG/2nn−1=en−12αWG−∑i=2nσiαG/2nn−1⋅e2n−1∑i=2nσiαG/2nn−1=e2n−1αWG+nσ2αG+,…,+σnαG/2nn−1.
Since(61)eσ1αG2n+2n−12nn−1∑i=1neσiαG−eσ1αG−Δ≥12n∑i=1neσiαG−12e2αWG/n,then(62)DEEαG=∑i=1neσiαG≥eσ1αG+2n−1Δ−n−1e2αWG/n.
Equality holds, that is,eσ1αG=eσ2αG=,…,=eσnαG if and only if G≅nK1.Lemma 8 (see [34]).
Fora1,a2,…,an≥0. Then,(63)n1n∑i=1nai−∏i=1nai1/n≤Ψ≤nn−1∑i=1nain−∏i=1nai1/n,where Ψ=n∑i=1nai−∑i=1nai2.Theorem 9.
LetG be a graph with n vertices. Then,(64)∑i=1neσiαG/22−ne2αWG/nn−1≤DEEαG≤∑i=1neσiαG/2−nn−1e2αWG/n.Proof.
Letai=eσiαG for i=1,2,…,n, by Lemma 8, we have(65)∑i=1neσiαG−n∏i=1neσiαG1/n≤Ψ≤nn−1∑i=1neσiαGn−∏i=1neσiαG1/n,where Ψ=n∑i=1neσiαG−∑i=1neσiαG/22.
Analyzing the left and right side of the previous inequality, respectively, by equation (5), we have(66)∑i=1neσiαG−n∏i=1neσiαG1/n≤n∑i=1neσiαG−∑i=1neσiαG/22n−1∑i=1neσiαG≥∑i=1neσiαG/22−ne2αWG/nDEEαG≥∑i=1neσiαG/22−ne2αWG/nn−1,n∑i=1neσiαG−∑i=1neσiαG/22≤nn−1∑i=1neσiαGn−∏i=1neσiαG1/nDEEαG≤∑i=1neσiαG/2−nn−1e2αWG/n.Also, the relation betweenα-distance Estrada index and α-distance energy are established.Theorem 10.
LetG be a graph with n vertices. Then,(67)DEEαG≤e2αWG/nn−1−ςαG+eςαG.Proof.
By the definition ofα-distance energy, we have(68)DEEαG=e2αWG/n∑i=1neσiαG−2αWG/n=e2αWG/nn+∑i=1n∑k=2∞σiαG−2αWG/nkk!≤e2αWG/nn+∑i=1n∑k=2∞σiαG−2αWG/nkk!≤e2αWG/nn+∑k=2∞1k!ςαGk=e2αWG/nn−1−ςαG+eςαG.


## 4. Conclusion

It is well-known that the graph energy and the Estrada index of graphs are important topics in graph theory. Theα-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.



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*Source: 9393521-2020-04-29.xml*