# New Bounds for the Generalized Distance Spectral Radius/Energy of Graphs

Authors: Yuzheng Ma; Yubin Gao; Yanling Shao
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

LetG be a simple connected graph with vertex set VG=v1,v2,…,vn and dvi be the degree of the vertex vi. Let DG be the distance matrix and TrG be the diagonal matrix of the vertex transmissions of G. The generalized distance matrix of G is defined as DαG=αTrG+1−αDG, where 0≤α≤1. If λ1,λ2,…,λn are the eigenvalues of DαG, then the generalized distance spectral radius of G is defined as ρDαG=max1≤i≤nλi. The generalized distance energy of G is EDαG=∑i=1n|λi−2αWG/n|, where WG is the Wiener index of G. In this paper, we give some bounds of the generalized distance spectral radius and the generalized distance energy.


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## Body

## 1. Introduction

Throughout this paper, we consider simple, connected, and finite graphs. LetG be such graph with vertex set VG=v1,v2,…,vn and edge set EG. Let dvi denote the degree of vertex vi and Nvi denote the neighbor set of vi. The distance between vertices vi and vj in G is the length of the shortest path connecting vi to vj, which is denoted as dvi,vj. The distance matrix [1, 2] of G is an n×n matrix DG=dij, where dij=dvi,vj for i,j=1,2,…,n.Definition 1.
LetG be a graph with vertex set VG=v1,v2,…,vn. The transmission of vertex vi, denoted by TrGvi or Tri, is defined to be the sum of the distances from vi to all vertices in G, that is, TrGvi=Tri=∑u∈VGdu,vi.The sequence Tr1,Tr2,…,Trn is the transmission degree sequence of G, and TrG=diagTr1,Tr2,…,Trn is the diagonal matrix of vertex transmissions of G.
Note that(1)
Transmission of a vertexv is also called the distance degree or the first distance degree of v.(2)
IfTrGvi=k for i=1,2,…,n, then G is called a k-transmission regular graph.Definition 2.
LetG be a graph with vertex setVG=v1,v2,…,vn, distance matrix DG, and transmission degree sequence Tr1,Tr2,…,Trn such that Tr1≥Tr2≥…≥Trn. Then, the second transmission of vertex vi, denoted by Ti, is defined to be Ti=∑j=1ndijTrj, and T1,T2,…,Tn is called the second transmission degree sequence of G.Definition 3.
(see [3]). Let G be a graph of order n. The Wiener index of G is defined as(1)WG=12∑i=1n∑j=1ndij=12∑vi,vj∈VGdij=12∑vi∈VGTrGvi.
In 1970, Gutman first proposed the concept of graph energy in [4]. The adjacency matrix AG of a graph G is a matrix of order n whose i,j-entry is equal to unity if the vertices vi and vj are adjacent and is equal to zero otherwise. Since AG is real and symmetric, all eigenvalues of AG are real, denoted by μ1,μ2,…,μn, also known as the eigenvalues of G. The energy of graph G is εG=∑i=1nμi. Let DegG=diagd1,d2,…,dn, where di=dvi for i=1,2,…,n. The matrices LG=DegG−AG and QG=DegG+AG are called the Laplacian matrix and the signless Laplacian matrix of graph G, respectively. For more research on Laplacian matrix and signless Laplacian matrix, refer to [5–8]. Aouchiche and Hansen [9, 10] introduced the distance Laplacian matrix DLG=TrG−DG and the distance signless Laplacian matrix DQG=TrG+DG of graph G.
In 2019, Cui et al. [11] proposed the generalized distance matrix DαG=αTrG+1−αDG by using the convex linear combination of TrG and DG, where 0≤α≤1. As you can see, D0G=DG, 2D1/2G=DQG, D1G=TrG, and DαG−DβG=α−βDLG. Since the matrix DαG is real and symmetric, all its eigenvalues are real, denoted by λ1,λ2,…,λn, which are called the generalized distance eigenvalues of G, and the generalized distance spectral radius of G is defined as ρDαG=max1≤i≤nλi.Definition 4.
(see [12]). Let G be a graph of order n. The generalized distance energy of G can be thought of as the mean deviation of the values of the generalized distance eigenvalues of G, namely, EDαG=∑i=1nλi−2αWG/2n.
The study of generalized distance spectrum was proposed by Cui et al. in [11]. They established some basic spectral properties of the generalized distance matrix of graphs, obtained the bounds of the generalized distance spectral radius, and determined the graphs with the minimum generalized distance spectral radius in all connected bipartite graphs with fixed vertices. In [13], the authors obtained some upper and lower bounds for the second largest eigenvalue of the generalized distance matrix of graphs, in terms of various graph parameters, and the graphs attaining the corresponding bounds are characterized. In [14], the authors obtained an upper bound for the smallest generalized distance eigenvalue λn in terms of different graph parameters. In particular, they showed that this upper bound is better than the upper bound obtained by Cui et al. In [14], the authors established the relations between the smallest eigenvalues of DαG, DG, and DQG and obtained sharp bounds for the smallest eigenvalue λn of DαG in terms of various graph parameters. In [12, 15–17], Alhevaz et al. established some new sharp bounds for the generalized distance spectral radius of G by using different graph parameters and characterized the extremal graphs. Then, they obtained new bounds for the k-th generalized distance eigenvalue. Moreover, through the eigenvalues of adjacency matrix and some auxiliary matrices, they studied the generalized distance spectrum of graphs obtained by generalization of the join graph operation. In 2020, they defined the generalized distance energy in [12] and gave the upper and lower bounds of the generalized distance energy. In [18, 19], Pirzada et al. obtained the bounds of the generalized distance spectral radius of bipartite graphs by using some parameters of graphs and characterized their extremal graphs. It was proved that for α∈1/2,1, the complete bipartite graph has the minimum generalized distance energy among all connected bipartite graphs, and for α∈0,2n/3n−2, the star has the minimum generalized distance energy among all trees. In addition, the generalized distance spectrum, generalized distance energy and the spectral spread of generalized distance matrix are also studied. For some recent results on spread of generalized distance matrix, we refer the readers to [20, 21] and the references therein. Inspired by the above literature, in this paper, we further study the generalized distance spectral radius and generalized distance energy.
The rest of the paper is organized as follows. In Section2, several lemmas are given. In Section 3, the new lower and upper bounds of the generalized distance spectral radius are obtained according to the distance between the vertices and some parameters of the graph. In Section 4, we obtain new bounds on the generalized distance energy in terms of spectral radius and parameters that depend on the distance between the vertices and the order of the graph.


## 2. Lemmas

In this section, we give some definitions and lemmas to prepare for subsequent proofs.Lemma 1 (see [22]).
IfA is a nonnegative real matrix of order n, then its spectral radius ρA is an eigenvalue of A and it has an associated nonnegative eigenvector. Furthermore, if A is irreducible, then ρA is a simple eigenvalue of A with an associated positive eigenvector.Lemma 2 (Rayleigh–Ritz theorem [23]).
IfB is a real symmetric matrix of order n with eigenvalues λ1≥λ2≥…≥λn, then for a nonzero vector X,(2)λ1≥XTBXXTX,with equality holding if and only if X is an eigenvector of B corresponding to λ1.Lemma 3 (Cauchy–Schwartz inequality).
Letak and bk be real numbers for all 1≤k≤n. Then,(3)∑k=1nakbk2≤∑k=1nak2∑k=1nbk2.Equality holds if and only ifajbk=akbj for all 1≤k,j≤n.Lemma 4 (see [11]).
LetG be a graph with distance degree sequence Tr1,Tr2,…,Trn. Then,(4)ρDαG≥2WGn.The equality holds if and only ifG is distance regular.Lemma 5 (see [11]).
LetG be a simple connected graph, Tri be the transmission of vertex vi, and Ti be the second transmission of vi. Then,(5)ρDαG≥∑i=1nαTri2+1−αTi2T1+T2+…+Tn.The equality holds if and only ifαTri+1−αTi/Tri is a constant for all i=1,2,…,n.Lemma 6 (see [24]).
Let the transmission degree sequence of graphG be Tr1,Tr2,…,Trn and the second transmission degree sequence of G be T1,T2,…,Tn. Then,(6)ρDαG≤max1≤i,j≤nαTri+Trj+α2Tri−Trj2+41−α2TiTj/TriTrj2,ρDαG≥min1≤i,j≤nαTri+Trj+α2Tri−Trj2+41−α2TiTj/TriTrj2.Each equality holds if and only ifG is a transmission regular graph.


## 3. Lower and Upper Bounds of Generalized Distance Spectral Radius

In this section, the matrix sequence is introduced according to the relationship between the transmission and the second transmission, and the bounds of the generalized distance spectral radius in Lemmas5 and 6 are generalized by using the matrix sequence in Theorems 1–3.Definition 5.
(see [25]). For i=1,2,…,n, the matrix sequence Mi1,Mi2,…,Mit… is defined as follows: fix β∈ℝ, let Mi1=Triβ, and for each t≥2, let Mit=∑j=1ndijMjt−1.
Note that forβ=1, Mi1=Tri, and Mi2=Ti for i=1,2,…,n.Theorem 1.
Let G be a connected graph of ordern, β be a real number, and t be an integer. Then,(7)ρDαG≥∑i=1nαTriMit+1−αMit+12∑i=1nMit2.The equality holds (for particular values ofβ and t) if and only if αTri+1−αMit+1/Mit is a constant for all i=1,2,…,n.Proof.
LetX=x1,x2,…,xnT be the unit positive Perron eigenvector of DαG corresponding to ρDαG. Let Y be the unit positive vector defined by(8)Y=1∑i=1nMit2M1t,M2t,…,MntT.
Note that(9)ρDαG=ρDαG2=XTDαG2X≥YTDαG2Y,(10)DαGY=1∑i=1nMit2αTr11−αd12⋯1−αd1n1−αd21αTr2⋯1−αd2n⋮⋮⋮1−αdn11−αdn2⋯αTrnM1tM2t⋮Mnt=1∑i=1nMit2αTr1M1t+1−αM2td12+…+Mntd1nαTr2M2t+1−αM1td21+…+Mntd2n⋮αTrnMnt+1−αM1tdn1+…+Mn−1tdnn−1=1∑i=1nMit2αTr1M1t+1−αM1t+1αTr2M2t+1−αM2t+1⋮αTrnMnt+1−αMnt+1.
We obtain(11)YTDαG2Y=∑i=1nαTriMit+1−αMit+12∑i=1nMit2.
Therefore,(12)ρDαG≥∑i=1nαTriMit+1−αMit+12∑i=1nMit2.
Now we assume that the equality holds in (7). By (9), Y is a positive eigenvector corresponding to ρDαG. From DαGY=ρDαGY, we obtain that αTri+1−αMit+1/Mit=ρDαG for i=1,2,…,n.
Conversely, ifαTri+1−αMit+1/Mit=k, then αTriMit+1−αMit+1=kMit for all i=1,2,…,n. Hence,(13)DαGY=1∑i=1nMit2kM1t,kM2t,…,kMntT=kY,that is, k is a eigenvalue of DαG and Y is a eigenvector corresponding to k. Because Y is a positive vector, applying Lemma 1, we obtain k=ρDαG, and(14)ρDαG=k=∑i=1nk2Mit2∑i=1nMit2=∑i=1nαTriMit+1−αMit+12∑i=1nMit2.
This completes the proof.Example 1.
Fori=1,…,6, let Gi be the graphs given in Figure 1. In particular, G4, G5, and G6 are the star, path, and cycle on seven vertices, denoted by S7, P7, and C7, respectively.
We observe thatG1 is a 7−transmission regular graph and G6 is a 12−transmission regular graph. In Table 1, we show the lower bounds for ρDαG, using four decimal places.Figure 1
Examples of six connected simple undirected graphs.Table 1
Lower bounds forρDαG, using four decimal places, where Gi is given in Figure 1.
G1G2G3G4G5G6ρDαG73.74039.762810.839718.657312Lemma473.59.428610.28571612Lemma573.64269.553210.756417.528512Theorem1 (β=1∧t=1)73.64269.553210.756417.528512Theorem1 (β=1∧t=2)73.62819.544010.745117.337412Theorem1 (β=1∧t=3)73.63049.546410.745717.338412Theorem1 (β=1∧t=4)73.63019.545510.745717.376612Theorem1 (β=2∧t=1)73.70809.638410.832818.251912Theorem1 (β=2∧t=2)73.61389.532910.738217.148712Theorem1 (β=2∧t=3)73.63269.549110.746117.410912Theorem1 (β=2∧t=4)73.63169.544510.745717.362412Theorem1 (β=−1∧t=1)73.39609.35599.518715.016712Theorem1 (β=−1∧t=2)73.65679.560810.773517.701812Theorem1 (β=−1∧t=3)73.62569.541910.744117.298712Theorem1 (β=−1∧t=4)73.63089.547110.745817.384212Theorem1 (β=−2∧t=1)73.59419.505910.644516.994412Theorem1 (β=−2∧t=2)73.63549.548810.750117.434512Theorem1 (β=−2∧t=3)73.62939.545210.745417.357812Theorem1 (β=−2∧t=4)73.63039.546010.745717.094412Theorem 2.
Let G be a connected graph of ordern and t be an integer. Then,(15)ρDαG≤max1≤i,j≤nαTri+Trj+α2Tri−Trj2+41−α2Mit+1Mjt+1/MitMjt2.
The equality holds if and only ift=1, β=1, and Mi1 is a constant for all i=1,2,…,n.Proof.
LetX=x1,x2,⋯,xnT be the unit positive Perron eigenvector of M−1DαGM corresponding to ρDαG, where(16)M=M1t0⋯00M2t⋯0⋮⋮⋱⋮00⋯Mnt.
Letxp=maxxi|i=1,2,…,n, xq=maxxi|xi≠xp,i=1,2,…,n, and the i,j-entry of matrix M−1DαGM be(17)αTri,i=j,1−αMjtMitdij,i≠j.
Therefore,(18)M−1DαGMX=ρDαGX.
Thep and q lines in (18) can be expressed by(19)ρDαG−αTrpxp=1−α∑i=1nMitMptdpixi≤1−αxqMpt∑i=1ndpiMit=1−αxqMptMpt+1,(20)ρDαG−αTrqxq=1−α∑i=1nMitMqtdqixi≤1−αxpMqt∑i=1ndqiMit=1−αxpMqtMqt+1.
Then,(21)ρDαG−αTrpρDαG−αTrqxpxq≤1−α2Mpt+1MptMqt+1Mqtxpxq,that is,(22)ρ2DαG−αTrp+TrqρDαG+α2TrpTrq−1−α2Mpt+1Mqt+1MptMqt≤0.
So,(23)ρDαG≤αTri+Trj+α2Tri−Trj2+41−α2Mit+1Mjt+1/MitMjt2.
Therefore,(24)ρDαG≤max1≤i,j≤nαTri+Trj+α2Tri−Trj2+41−α2Mit+1Mjt+1/MitMjt2.
Forβ=1, for i=1,2,…,n: Mi1=Tri,Mi2=Ti. Let Mi1=k. Then, Mi2=k2. We know that Tri=k and Ti=k2. According to Lemma 1, k is the largest eigenvalue of DαG, and ρDαG=k, so the equality holds in (15). On the contrary, if inequality (15) is equal, then x1=x2=…=xn can be obtained from (19) and (20), that is, ρDαG=αTr1+1−αM1t+1/M1t=αTr2+1−αM2t+1/M2t=…=αTrn+1−αMnt+1/Mnt=k. It means when t=1 and β=1, Mi1 is a constant for i=1,2,…,n. This completes the proof.Theorem 3.
LetG be a connected graph and t be an integer. Then,(25)ρDαG≥min1≤i,j≤nαTri+Trj+α2Tri−Trj2+41−α2Mit+1Mjt+1/MitMjt2.The equality holds if and only ift=1, β=1, and Mi1 is a constant for all i=1,2,…,n.Proof.
The proof method is the same as Theorem2.Example 2.
We consider the graphsG1,G2,G3,G4,G5,G6 given in Example 1 and bounds for ρDαG given in Lemma 6 and Theorems 1 and 2. Using four decimal places, we obtain the upper bounds for ρDαG, as shown in Table 2.Table 2
Bounds forρDαG, using four decimal places, where Gi is given in Figure 1.
G1G2G3G4G5G6Lemma6 (upper/lower bounds)7/73.9/3.133310.6545/9.088910.9091/7.000020/13.212/12Theorem1/2 (β=1∧t=1)7/73.9/3.133310.6545/9.088910.9091/7.000020/13.212/12Theorem1/2 (β=1∧t=2)7/73.9143/3.109110.7098/9.090710.9138/6.909120.1500/12.874112/12Theorem1/2 (β=1∧t=3)7/73.9120/3.112810.6908/9.093810.9135/6.913820.1203/12.935312/12Theorem1/2 (β=1∧t=4)7/73.9124/3.112210.6982/9.095010.9136/6.913520.1260/12.923112/12Theorem1/2 (β=2∧t=1)7/73.8250/3.311110.3669/9.192610.8595/8.833319.3524/15.166712/12Theorem1/2 (β=2∧t=2)7/73.9280/3.087810.7772/9.067410.9165/6.859520.3006/12.624412/12Theorem1/2 (β=2∧t=3)7/73.9099/3.116310.6686/9.085110.9134/6.916520.0934/12.992312/12Theorem1/2 (β=2∧t=4)7/73.9127/3.111710.7070/9.091110.9136/6.913420.1313/12.912012/12Theorem1/2 (β=−1∧t=1)7/74.1333/2.900011.4156/8.907311.1667/5.454522.5635/11.254912/12Theorem1/2 (β=−1∧t=2)7/73.8857/3.160010.6097/9.079510.9014/7.166619.8606/13.608112/12Theorem1/2 (β=−1∧t=3)7/73.9167/3.150310.7258/9.085310.9142/6.901420.1808/12.812812/12Theorem1/2 (β=−1∧t=4)7/73.9116/3.113410.6842/9.091210.9135/6.914220.1144/12.947912/12Theorem1/2 (β=−2∧t=1)7/73.9464/3.061910.8193/9.042710.9477/6.424020.4452/12.527512/12Theorem1/2 (β=−2∧t=2)7/73.9072/3.120910.6808/9.094610.9117/6.947720.0743/13.027012/12Theorem1/2 (β=−2∧t=3)7/73.9131/3.111010.7007/9.094410.9136/6.911720.1349/12.904912/12Theorem 4.
LetG be a graph of order n and size m. If the diameter of G is 2, then(26)ρDαG≥2n2−2m−2nn,with equality holding if and only if G is a transmission regular graph.Proof.
LetX=1,1,…,1T. Since ∑i=1nλi=∑i=1nTri, by Lemma 2, we have(27)ρDαG≥XTDαGXXTX=1n∑i=1nTri=1n∑i=1n2n−di−2=2n2−2m−2nn,that is, (26) holds.
IfG is a transmission regular graph, then Tr1=Tr2=…=Trn, and so the equality in (26) holds. Conversely, if the quality in (26) holds, it is clear that G is a transmission regular graph.


## 4. Upper Bound of Generalized Distance Energy

In this section, we obtain new bounds on the generalized distance energy in terms of spectral radius and parameters that depend on the distance between the vertices and the order of the graph. LetG be a graph of order n with generalized distance eigenvalues λ1,λ2,…,λn. Let θi=λi−2αWG/n for i=1,2,…,n. Then, θ1,θ2,…,θn are called the auxiliary eigenvalues of G [19]. It is easy to see that ∑i=1nθi=0, and(28)EDαG=∑i=1nλi−2αWGn=∑i=1nθi.Denote|θ|max=maxθ1|,|θ2|,…,|θn, |θ|min=minθ1|,|θ2|,…,|θn, and T=∑i=1nθi2.Theorem 5.
LetG be a connected graph of order n. Then,(29)EDαG≤θmax+n−1T−θmax2,ifθmax≤T/n,θmin+n−1T−θmin2,ifθmin≥T/n,Tn+n−1T−Tn,ifθmin≤T/n≤θmax.Proof.
Letλ1≥λ2≥…≥λnbe the generalized distance eigenvalues of G, and θi=λi−2αWG/n for i=1,2,…,n. By Lemma 3, for i=1,2,…,n, we get(30)EDαG−θi2≤n−1T−θi2,so(31)EDαG≤θi+n−1T−θi2.
We now consider the functionfx=x+n−1T−x2 with 0<x≤T. For T/n≤x≤T, f′x=1+−2nx+2x/2n−1T−x2<0,and fx is monotonically decreasing. For 0≤x≤T/n, f′x=1+−2nx+2x/2n−1T−x2>0,and fx is monotonically increasing. Thus, we have the following results.(i)
If|θ|max≤T/n, then EDαG≤f|θ|max=|θ|max+n−1T−|θ|max2.(ii)
If|θ|min≥T/n, then EDαG≤f|θ|min=|θ|min+n−1T−|θ|min2.(iii)
If|θ|min≤T/n≤|θ|max, then EDαG≤fT/n=T/n+n−1T−T/n.
This completes the proof.Lemma 7.
IfG is a graph of diameter 2, then(32)∑i=1nθi2=1−α24nn−1−6m+α2∑i=1ndi2−4m2n.Proof.
Since there are2m elements equal to 1 and nn−1−2m elements equal to 2 in DG, we have(33)∑i=1nθi2=1−α2∑i=1n∑j=1ndij2+α2∑i=1nTri2−αn∑i=1nTri2=1−α2∑i=1n∑j=1ndij2+α2∑i=1n2n−2−di2−1nα∑i=1n2n−2−di22=1−α22m+n2−n−2m⋅22+α24n−12n−8mn−1+∑i=1ndi2−α2n4n−12n2−8mnn−1+4m2=1−α24nn−1−6m+α2∑i=1ndi2−4m2n.
The lemma holds.Theorem 6.
IfG is a r-regular graph of diameter 2, then(34)EDαG≤1−α2n−2−r+n−11−α2r+22−n12−7r.Proof.
SinceG is a r-regular graph of diameter 2, by Theorem 4, ρDαG=2n2−2m−2n/n, and θ1=ρDαG−2αWG/n=2n−r−2−α2n−2−r=1−α2n−2−r. Applying Lemma 3 to 1,1,…,1T and θ2,θ3,…,θnT, we get(35)EDαG−θ12≤n−1∑i=2nθi2.
By Lemma7,(36)EDαG−θ12≤n−11−α24nn−1−6m+α2∑i=1ndi2−4m2n−θ12,that is,(37)EDαG≤θ1+n−11−α24nn−1−6m+α2∑i=1ndi2−4m2n−θ12.
Sinceθ1=1−α2n−2−r and 2m=nr, (37) gives(38)EDαG≤1−α2n−2−r+n−11−α2r+22−n12−7r.
The theorem follows.


## 5. Conclusions

In this paper, some new lower and upper bounds of the generalized distance spectral radius are obtained in terms of the distance between the vertices and some parameters of the graph. Meanwhile, we obtain new bounds on the generalized distance energy in terms of spectral radius and parameters that depend on the distance between the vertices and the order of the graph.



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