# A Note on Some Bounds of theα-Estrada Index of Graphs

Authors: Yang Yang; Lizhu Sun; Changjiang Bu
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

LetG be a simple graph with n vertices. Let A~αG=αDG+1−αAG, where 0≤α≤1 and AG and DG denote the adjacency matrix and degree matrix of G, respectively. EEαG=∑i=1neλi is called the α-Estrada index of G, where λ1,⋯,λn denote the eigenvalues of A~αG. In this paper, the upper and lower bounds for EEαG are given. Moreover, some relations between the α-Estrada index and α-energy are established.


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## Body

## 1. Introduction

LetG=VG,EG be a simple undirected graph with n vertices, where VG denotes the vertex set of G and EG denotes the edge set of G. Let AG and DG denote the adjacency matrix and degree matrix of G, respectively. The Laplacian (signless Laplacian) matrix of G is LG=DG−AGQG=DG+AG. In [1], Nikiforov defined A~αG=αDG+1−αAG0≤α≤1 and studied some problems of A~αG (for example, spectral extremal problems). Some results on A~αG have been obtained, including bounds of the eigenvalue of A~αG and the positive semidefiniteness of A~αG [1, 2], etc. For more spectral properties of A~αG, see [3–6].TheEstrada index [7] of G is defined as
(1)EEG=∑i=1neλi,where λ1,⋯,λn are eigenvalues of AG. The Estrada index can be used to measure the folding degree of long-chain proteins [8, 9] and subgraph centrality in complex networks [10–13].TheLaplacian Estrada index and signless Laplacian Estrada index of G are defined as LEEG=∑i=1neμi and SLEEG=∑i=1neqi, respectively, where μ1,⋯,μn and q1,⋯,qn are eigenvalues of LG and QG, respectively [14, 15]. Some mathematical and chemical properties of EEG, LEEG, and SLEEG are investigated extensively in mathematical chemistry [14, 16–25]. For other generalized Estrada index, see [26, 27].In [28], Guo and Zhou proposed the α-Estrada index as
(2)EEαG=∑i=1neλi=∑i=1n∑k=0∞λikk!,where λ1,⋯,λn are eigenvalues of A~αG. Obviously, EE0G is the Estrada index; note that EE1/2 is somewhat different from the signless Laplacian Estrada index, which is defined to be SLEEG=∑i=1ne2λi, where λi are the eigenvalues of A~1/2G.The paper is organized as follows: In Section2, some bounds for EEαG are obtained in terms of the number of vertices, edges, and triangles of G. We also give some new bounds for EEαG through different numerical inequalities. Furthermore, some relations between the α-Estrada index and α-energy are established. In Section 3, we compare our new bounds to the existing results for the α-Estrada index by certain graphs, benchmark graphs, and random graphs. In Section 4, we summarize the results of the paper, and the future work is envisaged.


## 2. Some Bounds for theα-Estrada Index

In what follows, lettrM denote the trace of matrix M. Let di denote the degree of vertex i.Lemma 1 (see [1, 5]).
LetG be a graph with m edges and t triangles. Then
(3)trAαG=2αm,(4)trAα2G=21−α2m+α2∑i∈VGdi2,(5)trAα3G=α3∑i∈VGdi3+3α1−α2∑i∈VGdi2+61−α3t.In this section, letτG=i,j,k⊂VG:i,j,kformatriangleofG; let β and ξ denote the numbers of subgraphs of G which are isomorphic to path P3 and cycle C4, respectively.Proposition 2.
LetG be a graph with m edges. Then
(6)trA~α4G=α4∑i∈VGdi4+4α21−α2∑i∈VGdi3+8α1−α3∑i,j,k∈τGdi+dj+dk+4α21−α2∑i,j∈EGdidj+1−α42m+4β+8ξ.Proof.
SincetrAB=trBA for any A,B∈ℝn×n, we have
(7)trDG2AG2=∑i∈VGdi3,trDGAG2=2∑i,j∈EGdidj,trDGAG3=2∑i,j,k∈τGdi+dj+dk.
It is known thattrA4G=2m+4β+8ξ (see [29]). Let A≔αDG and B≔1−αAG Taking the trace of A~α4G, we have
(8)trA+B4=trA+B2A+B2=trA2+AB+BA+B2A2+AB+BA+B2=trA4+4A2B2+4AB3+2AB2+B4=α4∑i∈VGdi4+4α21−α2∑i∈VGdi3+8α1−α3∑i,j,k∈τGdi+dj+dk+4α21−α2∑i,j∈EGdidj+1−α42m+4β+8ξ.In the following, we give a lower bound for theα-Estrada index of a graph by using the parameter α, the vertex number, the edge number, and the numbers of subgraphs of G.Theorem 3.
LetG be a graph with n vertices, m edges, and t triangles. Then
(9)EEαG≥n+2α+1−α2+1121−α4m+2α2+2α1−α2m2n+43α3+α21−α2·m3n2+α42m43n3+1−α3t+13α1−α3ζ+16α21−α2γ+161−α4β+2ξ,where γ=∑i,j∈EGdidj,ζ=∑i,j,k∈τGdi+dj+dk.Proof.
By definingEEαG, we have
(10)EEαG=∑k=0∞trA~αkGk!,EEαG≥n+trA~αG+trA~α2G2!+trA~α3G3!+trA~α4G4!.
According to the Hölder inequality, we have2m=∑i∈VGdi≤nt−1/t∑i∈VGdit1/t for any positive integer t. Hence
(11)∑i∈VGdit≥2mtnt−1.
By (3)–(11) and Proposition 2, we have
(12)EEαG≥n+1221−α2m+α2∑i∈VGdi2+16α3∑i∈VGdi3+3α1−α2∑i∈VGdi2+61−α3t+124α4∑i∈VGdi4+4α21−α2∑i∈VGdi3+8α1−α3∑i,j,k∈τGdi+dj+dk+4α21−α2γ+1−α42m+4β+8ξ+2αm≥n+2αm+1221−α2m+α24m2n+16α38m3n2+3α1−α24m2n+61−α3t+124α416m4n3+α21−α232m3n2+8α1−α3ζ+4α21−α2γ+1−α42m+4β+8ξ=n+2α+1−α2+1121−α4m+2α2+2α1−α2m2n+43α3+α21−α2m3n2+23α4m4n3+1−α3t+13α1−α3ζ+16α21−α2γ+161−α4β+2ξ,where γ=∑i,j∈EGdidj, ζ=∑i,j,k∈τGdi+dj+dk.Corollary 4.
LetG be an r-regular graph with n vertices and t triangles. Then
(13)EEαG>n+α+121−α2+1241−α4nr+12α2+α1−α2nr2+16α3+14α21−α2·nr3+124α4nr4+1−α3t+α1−α3rt+161−α4β+2ξ.Proof.
SinceG is an r-regular graph, then m=1/2nr, γ=∑ij∈EGdidj=1/2nr3, ζ=∑i,j,k∈τGdi+dj+dk=3rt. By Theorem 3, we have
(14)EEαG>n+α+121−α2+1241−α4nr+12α2+12α1−α2nr2+16α3+α21−α2nr3+124α4nr4+1−α3t+α1−α3rt+112α21−α2nr3+161−α4β+2ξ=n+α+121−α2+1241−α4nr+12α2+12α1−α2nr2+1−α3t+16α3+14α21−α2nr3+124α4nr4+α1−α3rt+161−α4β+2ξ.Also, we give another lower bound for theα-Estrada index of a graph including the parameter α, the vertex number, the edge number, and the numbers of triangles of G.Theorem 5.
LetG be a graph with n vertices, m edges, and t triangles. Then
(15)EEαG≥n2+8α2+8α1−α2m2+21−α2+4α2mn+α332m33n+21−α3nt.Proof.
Letλ1,⋯,λn be eigenvalues of A~αG. By the Taylor expansion theorem, then
(16)ex≥1+x+x22!+x33!,with equality if and only if x=0.
(17)EEαG2=∑i=1n∑j=1neλi+λj≥∑i=1n∑j=1n1+λi+λj+λi+λj22!+λi+λj33!=∑i=1n∑j=1n1+λi+λj+λiλj+λi22+λj22+λi2λj2+λiλj22+λi36+λj36.
By (3), we have
(18)∑i=1nλi=2αm.
So(19)∑i=1n∑j=1nλi+λj=n∑i=1nλi+n∑j=1nλj=4αmn,∑i=1n∑j=1nλiλj=∑i=1nλi2=4α2m2.
By (4) and (11), we have
(20)∑i=1nλi2≥21−α2m+α24m2n.
So(21)∑i=1n∑j=1nλi22+λj22=n2∑i=1nλi2+∑j=1nλj2≥21−α2mn+4α2m2,∑i=1n∑j=1nλi2λj2+λj2λi2=12∑i=1nλi2∑j=1nλj+∑i=1nλi∑j=1nλj2≥4α1−α2m2+8α3m3n.
By (5) and (11), we have
(22)∑i=1nλi3≥α38m3n2+3α1−α24m2n+61−α3t.
So(23)∑i=1n∑j=1nλi36+λj36=n3∑i=1nλi3≥21−α3nt+4α1−α2m2+8α3m33n.
Summarize the above conclusions, we have(24)EEαG2≥n2+4αmn+4α2m2+21−α2mn+4α2m2+4α1−α2m2+8α3m3n+21−α3nt+4α1−α2m2+8α3m33n=n2+8α2+8α1−α2m2+21−α2+4α2mn+α332m33n+21−α3nt.In order to prove Theorem8, we give two lemmas as follows:Lemma 6 (see [27]).
Letx1,x2,⋯,xn be nonnegative real numbers, and k>2, then
(25)∑i=1nxik≤∑i=1nxi2k/2.Lemma 7 (see [30]).
LetG be a graph with n vertices and m edges. Then
(26)∑i∈VGdi2≤m2mn−1+n−2.
Inspired by literatures [14, 18], we obtained some bounds on the α-Estrada index by arithmetic-geometric inequality.Theorem 8.
LetG be a graph with n vertices and m edges. Then
(27)n+4αm+nn−1e4am/n≤EEαG≤n+2αm−1−ω+eω,where ω=21−α2mn−1+2α2m2−α2n−1n−2m/n−1.Proof.
Letλ1,⋯,λn be eigenvalues of A~αG. Then
(28)EEαG2=∑i=1ne2λi+2∑i<jeλieλj.
By the arithmetic-geometric inequality, we have(29)2∑i<jeλieλj≥nn−1∏i<jeλieλj2/nn−1=nn−1∏i=1neλin−12/nn−1=nn−1e∑inλi2/n=nn−1e4am/n.
By the Taylor expansion theorem, we have(30)∑i=1ne2λi=∑i=1n∑k≥02λikk!=n+4αm+∑i=1n∑k≥22λikk!.
Letι∈0,4, we have
(31)∑i=1ne2λi≥n+4αm+ι∑i=1n∑k≥2λikk!=n+4αm−ιn−2αιm+ι∑i=1n∑k≥0λikk!=1−ιn+4α−2ιαm+ιEEαG.
By substituting the above formula and solving forEEαG, we obtain
(32)EEαG2≥1−ιn+4α−2ιαm+ιEEαG+nn−1e4am/n,EEαG≥ι2+1−ιn+4α−2ιαm+ι24+nn−1e4am/n.
It is elementary to show forn≥2; let the function
(33)fx=x2+1−xn+4α−2αxm+x24+nn−1e4am/n,where 0≤α≤1; then fx monotonically decrease in the interval 0,4. Let x=0; fx is max; that is to say, ι=0, EEαG is a better lower bound.
By Lemmas6 and 7, we have
(34)EEαG=n+2αm+∑i=1n∑k≥2λikk!≤n+2αm+∑i=1n∑k≥2λikk!=n+2αm+∑k≥21k!∑i=1nλik≤n+2αm+∑k≥21k!∑i=1nλi2k/2=n+2αm+∑k≥21k!21−α2m+α2∑i=1ndi2k/2≤n+2αm+∑k≥21k!21−α2m+α2m2mn−1+n−2k/2=n+2αm−1−ω+∑k≥01k!ωk=n+2αm−1−ω+eω,where ω=21−α2mn−1+2α2m2−α2n−1n−2m/n−1.In what follows, letλ1 and λn be the largest and the smallest of A~αG, respectively.Lemma 9 (see [1]).
LetG be a graph on n vertices with m edges. Then
(35)λ1≥2mn.The equality holds if and only if G is a regular graph.Theorem 10.
LetG be a graph on n vertices with m edges. Then
(36)EEαG≥e2m/n+n−1+2αm−2mn.Proof.
Consider the function(37)fx=x−1−lnx,x>0.
Obviously, the functionfx is decreasing in x∈0,1 and increasing in x∈1,+∞; then fx≥f1=0, implying that
(38)x≥1+lnx,x>0.The equality holds if and only if x=1. By Lemma 1, we have
(39)EEαG≥eλ1+n−1+∑k≥2lneλk=eλ1+n−1+∑k≥2λk=eλ1+n−1+2αm−λ1.
Define another function(40)Φx=ex+n−1+2αm−x,x>0.
Clearly, this is an increasing function onx∈0,+∞. On the other hand, by Lemma 9,
(41)λ1≥2mn≥0.Then,
(42)Φλ1≥Φ2mn.
Finally, we get(43)EEαG≥e2m/n+n−1+2αm−2mn.From Theorem10, we have the following result.Corollary 11.
LetG be a r-regular graph with n vertices. Then
(44)EEαG≥er+n−1+αnr−r.In the following, we also obtained some other bounds for theα-Estrada index through Sarasija’s [31], Ozeki’s [32], Polya’s [33], and Guo’s [34] inequalities, respectively.Lemma 12 (see [31]).
Letx1,x2,⋯,xn be nonnegative real numbers. Then
(45)n1n∑i=1nxi−∏i=1nxi1/n≤n∑i=1nxi−∑i=1nxi2≤nn−11n∑i=1nxi−∏i=1nxi1/n.Theorem 13.
LetG be a graph on n vertices with m edges. Then
(46)∑i=1neλi/22−ne2am/nn−1≤EEαG≤∑i=1neλi/22−nn−1e2am/n.Proof.
By Lemma 1 and Lemma12, let xi=eλi (i=1,2,⋯,n); we have
(47)n1n∑i=1neλi−∏i=1neλi1/n≤n∑i=1neλi−∑i=1neλi2≤nn−11n∑i=1neλi−∏i=1neλi1/n.Then
(48)∑i=1neλi−ne2am/n≤n∑i=1neλi−∑i=1neλi/22≤n−1∑i=1neλi−nn−1e2am/n.
Consider the left and right sides of inequality, respectively, we have(49)∑i=1neλi−ne2am/n≤n∑i=1neλi−∑i=1neλi/22,EEαG≥∑i=1neλi/22−ne2am/nn−1.
Similarly,(50)EEαG≤∑i=1neλi/22−nn−1e2am/n.Lemma 14 (see [32]).
Ifai and bi are positive real numbers for 1≤i≤n, then
(51)∑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 15.
LetG be a graph on n vertices with m edges. Then
(52)EEαG≥n∑i=1ne2λi−n24eλ1−eλn2.
Equality holds if and only ifG≅Kn¯.Proof.
Letai=eλi and bi=1; then m1=eλn, M1=eλ1, and m2=M2=1, respectively. According to Lemma 14, we have
(53)∑i=1ne2λi∑i=1n1−∑i=1neλi2≤n24eλ1−eλn2.Then
(54)EEα2G≥n∑i=1ne2λi−n24eλ1−eλn2,EEαG≥n∑i=1ne2λi−n24eλ1−eλn2.Lemma 16 (see [33]).
Supposeai and bi are positive real numbers for 1≤i≤n; then
(55)∑i=1nai2∑i=1nbi2≤14M1M2m1m2+m1m2M1M22∑i=1naibi2,where M1=max1≤i≤nai,M2=max1≤i≤nbi,m1=min1≤i≤nai,m2=min1≤i≤nbi.Theorem 17.
LetG be a graph on n vertices with m edges. Then
(56)EEαG≥2n∑i=1ne2λieλ1eλneλ1+eλn.Proof.
Letai=eλi and bi=1; then m1=eλn, M1=eλ1, and m2=M2=1, respectively. By Lemma 16, we have
(57)∑i=1ne2λi∑i=1n1≤14eλ1λn+eλnλ12∑i=1neλi2.Then
(58)EEαG≥2n∑i=1ne2λieλ1eλneλ1+eλn.Lemma 18 (see [34]).
Fora1,a2,⋯,an≥0 and p1,p2,⋯,pn≥0 such that ∑i=1npi=1. Then
(59)∑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 19.
LetG be a graph with n vertices with m edges. Then
(60)EEαG≥eλ1+2n−1Δ−n−1e2am/n,where Δ=e2αm2n−1−nλ1/2nn−1. Equality holds if and only if G≅Kn¯.Proof.
Letp1=1/2n, pi=2n−1/2nn−1 for i=2,⋯,n, ai=eλi and for i=2,⋯,n. Obviously, T=min1/2n,2n−1/2nn−1=1/2n, according to Lemma 18; we have
(61)eλ12n+2n−12nn−1∑i=2neλi−Δ≥121n∑i=1neλi−∏i=1neλi/n,where Δ=eλ1/2n∏i=2ne2n−1λi/2nn−1.
We considerΔ is
(62)Δ=eλ1/2n∏i=2ne2n−1λi/2nn−1=en−1λ1/2nn−1·e2n−1∑i=2nλi/2nn−1=en−1λ1/2nn−1·e2n−12αm−λ1/2nn−1=e2αm2n−1−nλ1/2nn−1.
Since(63)eλ12n+2n−12nn−1∑i=1neλi−eλ1−Δ≥12n∑i=1neλi−12e2am/n,
then(64)EEαG≥eλ1+2n−1Δ−n−1e2am/n,equality holds that is eλ1=eλ2=⋯=eλn if and only if G≅Kn¯.In the Hu¨ckel molecular orbital theory, graph energy is defined as the sum of the absolute values of the eigenvalues of the adjacency matrix of the molecular graph [35, 36]. In [28], the α-energy of G is defined as ςαG=∑i=1nλi−2am/n, where λ1,⋯,λn are the eigenvalues of A~αG. New bounds for the α-Estrada index in terms of the α-energy of the graph G are established.In [37], the Estrada index-like quantity is defined by
(65)EEG¯=∑i=1nexi−x¯,where x1,x2,⋯,xn are arbitrary real numbers and x¯ is their arithmetic mean. Let x1,x2,⋯,xn and x¯ be λ1,⋯,λn and 2am/n, respectively. Evidently, EEαG=e2am/nEEα¯G, and therefore, results obtained for EEαG can be immediately restated for EEα¯G and vice versa.Theorem 20.
LetG be a graph on n vertices with m edges. Then
(66)EEαG≤e2am/nn−1−ςαG+eςαG.Proof.
Note that∑i=1nλi−2am/n=0. By Lemma 9, we have
(67)e−2am/nEEαG=EEα¯G=n+∑i=1n∑k≥2λi−2am/nkk!≤n+∑i=1n∑k≥2λi−2am/nkk!≤n+∑k≥21k!ςαGk=n−1−ςαG+eςαG.
In order to prove Theorems21 and 22, let λi−2am/n=δii=1,⋯,n, and let δ1≥δ2≥⋯≥δk≥0>⋯≥δn. By Lemma 1, we obtained ∑i=1nδi=0. The α-energy of the graph G is ςαG=∑i=1nδi; then ςαG/2=∑δi≥0δi+=−∑δi<0δi−.In the following, some relations between theα-Estrada index and α-energy are established.Theorem 21.
LetG be a graph on n vertices with m edges. Then
(68)EEαG≥e2am/nςαG2−δ1+k−1+eδ1.Proof.
Letx≥0, considering the following function:
(69)fx=−1−x+ex,in which equality holds if and only ifx=0. The function fx is increasing in 0,+∞. The fx≥f0, implying that
(70)x≤ex−1,x≥0.
By (70), we have
(71)ςαG2=∑i=1,δi≥0kδi=δ1+∑i=2,δi≥0kδi≤δ1+∑i=2,δi≥0keδi−1=δ1−k−1+∑i=1,δi≥0keδi−eδ1≤δ1−k−1+∑i=1,δi≥0keδi+∑i=k+1,δi<0neδi−eδ1=δ1−k−1+∑i=1neδi−eδ1=δ1−k−1+EEα¯G−eδ1.Theorem 22.
LetG be a graph on n vertices with m edges. Then
(72)EEαG≥e2am/nkeζG/2k+n−ke−ζG/2n−k,in which equality holds if and only ifδ1=⋯=δk and δk+1=⋯=δn.Proof.
By the Mean Quadratic inequality, we have(73)∑i=1,δi≥0keδi≥keδ1+δ2+⋯+δk/k=keζG/2k.
Similarly,(74)∑i=k+1,δi<0neδi≥n−ke−ζG/2n−k.
Then(75)EEαG=e2am/nEE¯αG≥e2am/nkeζG/2k+n−ke−ζG/2n−k.The equality holds in (75) if and only if equalities hold in both (73) and (74). By the equality case in the Mean Quadratic inequality, equality occurs in (73) and (74) if and only if eδ1=⋯=eδk and eδk=⋯=eδn; that is to say, the equality holds in (75) if and only if δ1=⋯=δk and δk+1=⋯=δn. This means all negative eigenvalues and all nonnegative eigenvalues which completes the proof.


## 3. Numerical Examples

In this section, we list some computational experiments to compare our new bounds to previous results for certain connected graphs, benchmark graphs, and random graphs, where the results of the benchmark graphs and random graphs are the average of 20 independent experiments. We listed the lower bound of Theorem 1 (Th. 1) [23], the lower bound of Theorem 10 (Th. 10), the lower bound of Theorem 13 (Th. 13–), the upper bound of Theorem 2.1 (Th. 2.1) [38], the upper bound of Theorem 13 (Th. 13+), and the numerical value of EE0G (see Table 1).Table 1
GraphG
EE0G
Th. 1
Th. 10
Th. 13–
Th. 2.1
Th. 13+
P3
5.3563
4.1231
4.4603
4.6993
7.6522
6.3987
P4
7.6357
5.2915
5.9816
6.3059
12.3564
10.9177
P5
9.9153
6.4031
7.3530
7.9106
18.0940
16.6426
P6
12.1949
7.4833
8.6278
9.5146
25.0855
23.5733
S1,3
7.8291
5.2915
5.9816
6.3404
12.3564
11.0214
S1,4
10.5243
6.4031
7.3530
8.0103
18.0940
17.0413
S1,5
13.4633
7.4833
8.6278
9.7098
25.0855
24.5493
C4
9.5243
5.6569
8.3890
7.2896
17.0940
13.8690
C5
11.4961
6.7082
9.3890
8.7768
24.0855
20.1074
C6
13.6967
7.7459
10.3890
10.3418
32.5671
27.7093
K2,3
14.6687
7
12.6231
9.9628
31.5671
24.8515
K3,3
24.1353
8.4853
22.0855
13.9547
66.7507
45.7738
K4
21.1891
6.3245
20.0855
10.6829
30.5671
27.7061
K5
56.0696
8.0622
54.5981
22.8344
82.1710
76.3377
K6
150.2526
9.7979
148.4132
45.1001
223.1460
201.5007
Petersen
34.2182
13.4164
26.0855
21.2832
227.1460
111.5495
C20
66.6083
22.8035
36.0855
41.2607
2.1861f3
423.9540
C40
132.0312
42.8952
56.0855
81.3404
5.4689f4
1.6522f3
C60
197.4502
62.9285
76.0855
121.4400
6.4640f5
3.6849f3
ER1
1.2052f3
105.2235
306.3649
378.3391
1.1098f10
2.7655f4
ER2
8.7546f21
141.4779
5.6165f21
8.8431f19
5.3795f30
8.7546f21
BA
1.9931f11
116.1895
3.9824f7
2.0157f9
1.4542f18
1.9956f11
WS1
2.4416f3
105.8304
496.4287
625.7075
4.2572f10
5.2145f4
WS2
1.5298f3
105.8301
496.5374
457.8648
4.2572f10
3.5528f4
GN
1.4226f3
129.9692
132.2901
281.0178
8.0887f6
1.9561f4
LFR
5.5072f6
1.0093f3
1.3421f4
3.2617f3
1.4683f13
9.3818f7TheC20, C40, and C60 are fullerenes (letter C is followed by the number of carbon atoms). ER1 is the Erdös-Rényi random graph with n=100 and p=0.05. ER2 is the Erdös-Rényi random graph with n=100 and p=0.5. BA is the Barabási-Albert random graph with n=100, m=5, and n0=50. WS1 is the Watts-Strogatz random graph with n=100, K=6, and p=0.1. WS2 is the Watts-Strogatz random graph with n=100 and K=6, and p=0.5. GN is the GN (Girvan-Newman) Benchmark graph with n=128, k=16, maxk=16, mu=0.1, minc=32, and maxc=32. LFR is the LFR (Lancichinetti-Fortunato-Radicchi) Benchmark graph with n=1000, k=10, maxk=40, mu=0.2, minc=30, and maxc=60 (for related parameters, see [39, 40]). We use fx instead of 10x in Table 1. The results are kept to four decimal places.According to the information in Table1, we know that the lower bounds in Th. 2.10 and Th. 2.13− are better than the lower bound of Th. 1; the upper bound of Th. 2.13+ is better than the upper bound of Th. 2.1. We also get some other results in Table 1 as follows: The lower bound of Th. 2.10 is better than the lower bounds of Th. 2.13− in the cycle graph, bipartite graph, Petersen graph, ER2, WS2, and LFR; Th. 2.13− is good in other cases. For sparse graphs, Th. 2.13− is good in most cases. For dense graph, Th. 2.10 is good in most cases.


## 4. Conclusion

In this paper, we give some bounds on theα-Estrada index of G, some relations between the α-Estrada index and α-energy are established. At the same time, we also analyze the advantages and disadvantages of different bounds for certain connected graphs, benchmark graphs, and random graphs by numerical experiments. Our future work will focus on exploring the practical applications of the α-Estrada index in physical, chemical, and network sciences.



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*Source: 3972789-2020-03-16.xml*