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# On the distance *α*-spectral radius of a connected graph

*Journal of Inequalities and Applications*
**volume 2020**, Article number: 161 (2020)

## Abstract

For a connected graph *G* and \(\alpha \in [0,1)\), the distance *α*-spectral radius of *G* is the spectral radius of the matrix \(D_{\alpha }(G)\) defined as \(D_{\alpha }(G)=\alpha T(G)+(1-\alpha )D(G)\), where \(T(G)\) is a diagonal matrix of vertex transmissions of *G* and \(D(G)\) is the distance matrix of *G*. We give bounds for the distance *α*-spectral radius, especially for graphs that are not transmission regular, propose local graft transformations that decrease or increase the distance *α*-spectral radius, and determine the graphs that minimize and maximize the distance *α*-spectral radius among several families of graphs.

## 1 Introduction

We consider simple and undirected graphs. Let *G* be a connected graph of order *n* with vertex set \(V(G)\) and edge set \(E(G)\). For \(u,v\in V(G)\), the distance between *u* and *v* in *G*, denoted by \(d_{G}(u,v)\) or simply \(d_{uv}\) if the graph *G* is clear from the context, is the length of a shortest path from *u* to *v* in *G*. The distance matrix of *G* is the \(n\times n\) matrix \(D(G)=(d_{G}(u,v))_{u,v\in V(G)}\). For \(u\in V(G)\), the transmission of *u* in *G*, denoted by \(T_{G}(u)\), is defined as the sum of distances from *u* to all other vertices of *G*, i.e., \(T_{G}(u)=\sum_{v\in V(G)}d_{G}(u,v)\). The transmission matrix \(T(G)\) of *G* is the diagonal matrix of transmissions of *G*. Then \(Q(G)=T(G)+D(G)\) is the distance signless Laplacian matrix of *G*, proposed recently in [1]. Arisen from a data communication problem, the spectrum of the distance matrix was studied by Graham and Pollack [12] in 1971, early related work may be found also in [10, 11], and now it has been studied extensively, see the recent survey [2] and the very recent papers [4, 5, 17, 18, 26]. The distance signless Laplacian spectrum has also received much attention, see, e.g., [1, 3, 4, 7, 15, 16, 29].

Throughout this paper we assume that \(\alpha \in [0,1)\). Motivated by the work of Nikiforov [22], we consider the convex combinations \(D_{\alpha }(G)\) of \(T(G)\) and \(D(G)\), defined as

see [6]. Evidently, \(D_{0}(G)=D(G)\) and \(2D_{1/2}(G)=Q(G)\). We call the eigenvalues of \(D_{\alpha }(G)\) the distance *α*-eigenvalues of *G*. As \(D_{\alpha }(G)\) is a symmetric matrix, the distance *α*-eigenvalues of *G* are all real, which are denoted by \(\mu ^{(1)}_{\alpha } (G), \ldots , \mu ^{(n)}_{\alpha }(G)\), arranged in nonincreasing order, where \(n=|V(G)|\). The largest distance *α*-eigenvalue \(\mu ^{(1)}_{\alpha } (G)\) of *G* is called the distance *α*-spectral radius of *G*, written as \(\mu _{\alpha } (G)\). Obviously, \(\mu ^{(1)}_{0} (G), \ldots , \mu ^{(n)}_{0}(G)\) are the distance eigenvalues of *G*, and \(2\mu ^{(1)}_{1/2} (G), \ldots , 2\mu ^{(n)}_{1/2}(G)\) are the distance signless Laplacian eigenvalues of *G*. Particularly, \(\mu _{0}(G)\) is just the distance spectral radius [2] and \(2\mu _{1/2}(G)\) is just the distance signless Laplacian spectral radius of *G* [1].

In this paper, we give sharp bounds for the distance *α*-spectral radius, and particularly an upper bound for the distance *α*-spectral radius of connected graphs that are not transmission regular, and propose some types of graft transformations that decrease or increase the distance *α*-spectral radius. We also determine the unique graphs with minimum distance *α*-spectral radius among trees and unicyclic graphs, respectively, as well as the unique graphs (trees) with maximum and second maximum distance *α*-spectral radii, and the unique graph with maximum distance *α*-spectral radius among connected graphs with given clique number, and among odd-cycle unicyclic graphs, respectively.

## 2 Preliminaries

Let *G* be a connected graph with \(V(G)=\{v_{1},\ldots ,v_{n}\}\). A column vector \(x=(x_{v_{1}},\ldots , x_{v_{n}})^{\top }\in \mathbb{R}^{n}\) can be considered as a function defined on \(V(G)\) which maps vertex \(v_{i}\) to \(x_{v_{i}}\), i.e., \(x(v_{i})=x_{v_{i}}\) for \(i=1,\ldots ,n\). Then

or equivalently,

Since \(D_{\alpha }(G) \) is a nonnegative irreducible matrix, by the Perron–Frobenius theorem, \(\mu _{\alpha } (G)\) is simple and there is a unique positive unit eigenvector corresponding to \(\mu _{\alpha } (G)\), which is called the distance *α*-Perron vector of *G*. If *x* is the distance *α*-Perron vector of *G*, then for each \(u\in V(G)\),

which is called the *α*-equation of *G* at *u*. For a unit column vector \(x\in \mathbb{R}^{n}\) with at least one nonnegative entry, by Rayleigh’s principle, we have \(\mu _{\alpha } (G)\ge x^{\top }D_{\alpha }(G)x\) with equality if and only if *x* is the distance *α*-Perron vector of *G*.

As in [27], we have the following result.

### Lemma 2.1

*Suppose that**G**is a connected graph*, *η**is an automorphism of**G*, *and**x**is the distance**α*-*Perron vector of**G*. *Then for*\(u,v\in V(G)\), \(\eta (u)=v\)*implies that*\(x_{u}=x_{v}\).

### Proof

Let \(P=(p_{uv})_{u,v\in V(G)}\) be the permutation matrix such that \(p_{vu}=1\) if and only if \(\eta (u)=v\) for \(u,v\in V(G)\). We have \(D_{\alpha }(G) = P^{\top }D_{\alpha }(G) P\) and *Px* is a positive unit vector. Thus \(\mu _{\alpha }(G)=x^{\top }D_{\alpha }(G)x=(Px)^{\top }D_{\alpha }(G)(Px)\), implying *Px* is also the distance *α*-Perron vector of *G*. Thus \(P x = x\), and the result follows. □

Let *G* be a graph. For \(v\in V(G)\), let \(N_{G}(v)\) be the set of neighbors of *v* in *G*, and \(\operatorname{deg}_{G}(v)\) be the degree of *v* in *G*. Let \(G-v\) be the subgraph of *G* obtained by deleting *v* and all edges containing *v*. For \(S\subseteq V(G)\), let \(G[S]\) be the subgraph of *G* induced by *S*. For a subset \(E'\) of \(E(G)\), \(G-E'\) denotes the graph obtained from *G* by deleting all the edges in \(E'\), and in particular, we write \(G-xy\) instead of \(G-\{xy\}\) if \(E'=\{xy\}\). Let *G̅* be the complement of *G*. For a subset \(E'\) of \(E(\overline{G})\), denote \(G+E'\) the graph obtained from *G* by adding all edges in \(E'\), and in particular, we write \(G+xy\) instead of \(G+\{xy\}\) if \(E'=\{xy\}\).

For a nonnegative square matrix *A*, the Perron–Frobenius theorem implies that *A* has an eigenvalue that is equal the maximum modulus of all its eigenvalues; this eigenvalue is called the spectral radius of *A*, denoted by \(\rho (A)\). Note that \(\mu _{\alpha }(G)=\rho (D_{\alpha }(G))\) for a connected graph *G*.

Restating Corollary 2.2 in [20, p. 38], we have

### Lemma 2.2

([20])

*Suppose that**A**and**B**are square nonnegative matrices*, *A**is irreducible*, *and*\(A-B\)*is nonnegative but nonzero*. *Then*\(\rho (A)> \rho (B)\).

By Lemma 2.2, we have

### Lemma 2.3

*Suppose that**G**is a connected graph with*\(u,v\in V(G)\), *and**u**and**v**are not adjacent*. *Then*\(\mu _{\alpha }(G+uv)< \mu _{\alpha }(G)\).

The transmission of a connected graph *G*, denoted by \(\sigma (G)\), is the sum of distances between all unordered pairs of vertices in *G*. Clearly, \(\sigma (G)=\frac{1}{2}\sum_{v\in V(G)} T_{G}(v)\). A graph is said to be transmission regular if \(T_{G}(v)\) is a constant for each \(v\in V(G)\). By Rayleigh’s principle, we have

### Lemma 2.4

*Suppose that**G**is a connected graph of order**n*. *Then*\(\mu _{\alpha }(G)\ge \frac{2\sigma (G)}{n}\)*with equality if and only if**G**is transmission regular*.

For an \(n\times n\) nonnegative matrix \(A=(a_{ij})\), let \(r_{i}\) be the *i*th row sum of *A*, i.e., \(r_{i}=\sum_{j=1}^{n} a_{ij}\) for \(i=1, \ldots , n\), and let \(r_{\min }\) and \(r_{\max }\) be the minimum and maximum row sums of *A*, respectively.

### Lemma 2.5

([3])

*Let*\(A=(a_{ij})\)*be an*\(n\times n\)*nonnegative matrix with row sums*\(r_{1},\ldots ,r_{n}\). *Let*\(S=\{1,\ldots ,n\}\), \(r_{\min }=r_{p}\), \(r_{\max }=r_{q}\)*for some**p**and**q**with*\(1 \le p\), \(q\le n \), \(\ell =\max \{r_{i}-a_{ip}: i\in S\setminus \{p\}\}\), \(m=\min \{r_{i}-a_{iq}: i\in S\setminus \{q\}\}\), \(s=\max \{a_{ip}: i\in S\setminus \{p\}\}\)*and*\(t=\min \{a_{iq}: i\in S\setminus \{q\}\}\). *Then*

*Moreover*, *the first equality holds if*\(r_{i}-a_{iq}=m\)*and*\(a_{iq}=t\)*for all*\(i\in S\setminus \{q\}\), *and the second equality holds if*\(r_{i}-a_{ip}=\ell \)*and*\(a_{ip}=s\)*for all*\(i\in S\setminus \{p\}\).

Let \(J_{s\times t}\) be the \(s\times t\) matrix of all 1’s, \(0_{s\times t}\) the \(s\times t\) matrix of all 0’s, and \(I_{s}\) the identity matrix of order *s*.

Let \(K_{n}\), \(P_{n}\), and \(S_{n}\) be the complete graph, the path, and the star of order *n*, respectively. Let \(C_{n}\) denote the cycle of order \(n\ge 3\).

For a connected graph *G*, let \(T_{\min }(G)\) and \(T_{\max }(G)\) be the minimum and maximum transmissions of *G*, respectively.

## 3 Bounds for the distance *α*-spectral radius

Let *G* be a connected graph of order *n*. Note that \(D_{\alpha }(K_{n})=\alpha (n-1) I_{n}+(1-\alpha )(J_{n\times n}-I_{n})\), and thus \(\mu _{\alpha }(K_{n})=n-1\). By Lemma 2.3, we have \(\mu _{\alpha }(G)\ge n-1\) with equality if and only if \(G\cong K_{n}\).

If \((d_{1},\ldots , d_{n})\) is the nonincreasing degree sequence of a graph *G* of order at least 2, then \(d_{1}\) (resp. \(d_{2}\)) is the maximum (resp. second maximum) degree, \(d_{n}\) (resp. \(d_{n-1}\)) is the minimum (resp. second minimum) degree of *G*. The diameter of *G* is the maximum distance between all vertex pairs of *G*. Using techniques from [33] by considering the first two minima or maxima of the entries of the distance *α*-Perron vector, we may prove the following lower and upper bounds: If *G* is a connected graph of order \(n\ge 2\) with maximum degree Δ and second maximum degree \(\Delta '\), then

with equality if and only if *G* is regular with diameter at most 2. If *G* is a connected graph of order \(n\ge 2\) with minimum degree *δ* and second minimum degree \(\delta '\), then

with equality if and only if *G* is regular with \(d\le 2\), where *d* is the diameter of *G*, \(S=dn-\frac{d(d-1)}{2}-1-\delta (d-1)\) and \(S'=dn-\frac{d(d-1)}{2}-1-\delta ' (d-1)\). The proof of the above bounds may be found in the early version of this paper at arXiv:1901.10180.

Similarly, bounds for the distance *α*-spectral radius for connected bipartite graphs may be obtained as in [33].

A connected graph *G* of order *n* is distinguished vertex deleted regular (DVDR) if there is a vertex *v* of degree \(n-1\) such that \(G-v\) is regular. By the techniques in [3], we have the following bounds. For completeness, we include a proof here.

### Theorem 3.1

*Let**G**be a connected graph and**u**and**v**be vertices such that*\(T_{G}(u)=T_{\min }(G)\)*and*\(T_{G}(v)=T_{\max }(G)\). *Let*\(m_{1}=\max \{T_{G}(w)-(1-\alpha )d(u,w): w\in V(G)\setminus \{u\}\}\), \(m_{2}=\min \{T_{G}(w)-(1-\alpha )d(v,w): w\in V(G)\setminus \{v\}\}\), *and*\(e(w)=\max \{d(w,z): z\in V(G)\}\)*for*\(w\in V(G)\). *Then*

*The first equality holds if and only if**G**is a complete graph and the second equality holds if and only if**G**is a DVDR graph*.

### Proof

Let *M* be the submatrix of \(D_{\alpha }(G)\) obtained by deleting the row and column corresponding to vertex *v*. Let \(M'\) be the matrix obtained from *M* by reducing some nondiagonal entries of each row with row sum greater than \(m_{2}\) in *M* such that \(M'\) is nonnegative and each row sum in \(M'\) is \(m_{2}\).

Let \(D^{(1)}\) be the matrix obtained from \(D_{\alpha }(G)\) by replacing all \((w,v)\)-entries by \(1-\alpha \) for \(w\in V(G)\setminus \{v\}\), and replacing the submatrix *M* by \(M'\). Obviously, \(D_{\alpha }(G)\) and \(D^{(1)}\) are nonnegative and irreducible, and \(D_{\alpha }(G)\ge D^{(1)}\). By Lemma 2.2, we have \(\mu _{\alpha }(G)\ge \rho (D^{(1)})\) with equality if and only if \(D_{\alpha }(G)=D^{(1)}\). By applying Lemma 2.5 to \(D^{(1)}\), we obtain the lower bound for \(\mu _{\alpha }(G)\). Suppose that this lower bound is attained. Then \(D_{\alpha }(G)=D^{(1)}\). As all \((w,v)\)-entries are equal to \(1-\alpha \) for \(w\in V(G)\setminus \{v\}\), implying \(\operatorname{deg}_{G}(v)=n-1\). As \(T_{G}(v)=T_{\max }(G)\), *G* is a complete graph. Conversely, if *G* is a complete graph, then it is obvious that the lower bound for \(\mu _{\alpha }(G)\) is attained.

Let *C* be the submatrix of \(D_{\alpha }(G)\) obtained by deleting the row and column corresponding to vertex *u*. Let \(C'\) be the matrix obtained from *C* by adding positive numbers to nondiagonal entries of each row with row sum less than \(m_{1}\) in *C* such that each row sum in \(C'\) is \(m_{1}\). Let \(D^{(2)}\) be the matrix obtained from \(D_{\alpha }(G)\) by replacing all \((w,u)\)-entries by \((1-\alpha )e(u)\) for \(w\in V(G)\setminus \{u\}\), and replacing the submatrix *C* by \(C'\). Note that \(D_{\alpha }(G)\) and \(D^{(2)}\) are nonnegative and irreducible, and \(D^{(2)}\ge D_{\alpha }(G)\). By Lemma 2.2, \(\mu _{\alpha }(G)\le \rho (D^{(2)})\) with equality if and only if \(D_{\alpha }(G)=D^{(2)}\). By applying Lemma 2.5 to \(D^{(2)}\), we obtain the upper bound for \(\mu _{\alpha }(G)\).

Suppose that this upper bound is attained. By Lemma 2.2, \(D_{\alpha }(G)=D^{(2)}\). As all \((w,u)\)-entries are equal to \((1-\alpha )e(u)\) for \(w\in V(G)\setminus \{u\}\), implying \(e(u)=1\), i.e., \(\operatorname{deg}_{G}(u)=n-1\). Note that \(T_{G}(w)=m_{1}+1-\alpha \) for all \(w\in V(G)\setminus \{u\}\) and \(T_{\min }(G)=T_{G}(u)=n-1\). If \(m_{1}+1-\alpha =n-1\), then *G* is a complete graph, which is a DVDR graph. Otherwise, \(m_{1}+1-\alpha >n-1\).

Recall from [3] that an incomplete connected graph of order *n* is a DVDR graph if and only if except one vertex of degree \(n-1\) each other vertex has the same transmission. Thus, the upper bound for \(\mu _{\alpha }(G)\) is attained if and only if *G* is a DVDR graph. □

We mention that more bounds for \(\mu _{\alpha }(G)\) may be derived even from some known bounds for nonnegative matrices, see, e.g., [9].

Let *G* be a connected graph of order *n*. Let \(\varLambda =T_{\max }(G)\). As \(\mu _{\alpha }(G)\le \varLambda \) with equality if and only if *G* is transmission regular. For a connected non-transmission-regular graph *G* of order *n*, Liu et al. [19] showed that

and

Note that \(4\sigma (G)< n^{2}\varLambda \). We show new bounds as follows:

and

Instead of proving the two inequalities, we prove the following somewhat general result.

### Theorem 3.2

*Let**G**be a connected non*-*transmission*-*regular graph of order**n*. *Then*

*where*\(\varLambda =T_{\max }(G)\).

### Proof

Let *x* be the *α*-Perron vector of *G*. Denote by \(x_{u}=\max \{x_{w}: w\in V(G)\}\) and \(x_{v}=\min \{x_{w}: w\in V(G)\}\). Since *G* is not transmission regular, we have \(x_{u}> x_{v}\), and thus

implying that \(x_{u}^{2}>\frac{\mu _{\alpha }(G)}{2\sigma (G)}\). Note that

We need to estimate \(\sum_{\{w,z\}\subseteq V(G)}d_{wz}(x_{w}-x_{z})^{2}\). Let \(P=w_{0}w_{1}\ldots w_{\ell}\) be a shortest path connecting *u* and *v*, where \(w_{0}=u\), \(w_{\ell}=v\), and \(\ell\ge 1\). Obviously,

where \(N_{1}=\sum_{w\in V(G)\setminus V(P)}\sum_{z\in V(P)}d_{wz}(x_{w}-x_{z})^{2}\) and \(N_{2}=\sum_{\{w,z\}\subseteq V(P)}d_{wz}(x_{w}-x_{z})^{2}\). For \(w\in V(G)\setminus V(P)\), by the Cauchy–Schwarz inequality, we have

and thus

For \(1\le i \le \ell -1\) and \(\ell \ge 2\), by the Cauchy–Schwarz inequality, we have

and thus

*Case 1.**u* and *v* are adjacent, i.e., \(\ell =1\).

In this case, we have

Thus

Viewed as a function of \(x_{v}\), \((n\varLambda -2\sigma (G))x_{v}^{2}+(1-\alpha )\frac{n}{2}(x_{u}-x_{v})^{2}\) achieves its minimum value \(\frac{(1-\alpha )n(n\varLambda -2\sigma (G))}{2(n\varLambda -2\sigma (G))+(1-\alpha )n}x_{u}^{2}\). Recall that \(x_{u}^{2}>\frac{\mu _{\alpha }(G)}{2\sigma (G)}\). Then we have

which implies that

*Case 2.**u* and *v* are not adjacent, i.e., \(\ell \ge 2\).

Suppose first that *ℓ* is even. Then

Thus

Viewed as a function of \(x_{v}\), \((n\varLambda -2\sigma (G))x_{v}^{2}+(1-\alpha ) \frac{\ell ^{2}+4\ell +4n-4}{8}(x_{u}-x_{v})^{2}\) achieves its minimum value \(\frac{(1-\alpha )(n\varLambda -2\sigma (G))(\ell ^{2}+4\ell +4n-4)}{8(n\varLambda -2\sigma (G))+(1-\alpha )(\ell ^{2}+4\ell +4n-4)}x_{u}^{2}\). As \(x_{u}^{2}>\frac{\mu _{\alpha }(G)}{2\sigma (G)}\), we have

i.e.,

As a function of *ℓ*, the expression on the right-hand side in the above inequality is strictly increasing for \(\ell \ge 2\). Thus we have

Now suppose that *ℓ* is odd. Then

Thus, as early, we have

implying

As a function of *ℓ*, the expression on the right-hand side in the above inequality is strictly increasing for \(\ell \ge 3\). Thus we have

The result follows by combining Cases 1 and 2. □

## 4 Effect of graft transformations on distance *α*-spectral radius

In this section, we study the effect of some local graft transformations on distance *α*-spectral radius.

A path \(u_{0}\cdots u_{r}\) (with \(r\geq 1\)) in a graph *G* is called a pendant path (of length *r*) at \(u_{0}\) if \(\operatorname{deg}_{G}(u_{0})\geq 3\), the degrees of \(u_{1},\ldots ,u_{r-1}\) (if any exists) are all equal to 2 in *G*, and \(\operatorname{deg}_{G}(u_{r})=1\). A pendant path of length 1 at \(u_{0}\) is called a pendant edge at \(u_{0}\).

A vertex of a graph is a pendant vertex if its degree is 1. A cut edge of a connected graph is an edge whose removal yields a disconnected graph.

If *P* is a pendant path of *G* at *u* with length \(r\geq 1\), then we say *G* is obtained from *H* by attaching a pendant path *P* of length *r* at *u* with \(H=G[V(G)\setminus (V(P)\setminus \{u\})]\). If the pendant path of length 1 is attached to a vertex *u* of *H*, then we also say that a pendant vertex is attached to *u*.

### Theorem 4.1

*Suppose that**G**is a connected graph*, *uv**is a cut edge with*\(\operatorname{deg}_{G}(u)\ge 2\), *and**v**is adjacent to a pendant vertex*\(v'\). *Let*

*Then*\(\mu _{\alpha }(G)>\mu _{\alpha }(G_{uv})\).

### Proof

Let \(G_{1}\) and \(G_{2}\) be the components of \(G-uv\) containing *u* and *v*, respectively. Let *x* be the distance *α*-Perron vector of \(G_{uv}\). By Lemma 2.1, \(x_{u}=x_{v'}\). As we pass from *G* to \(G_{uv}\), the distance between a vertex in \(V(G_{1})\setminus \{u\}\) and a vertex in \(V(G_{2})\) is decreased by 1, the distance between a vertex \(V(G_{1})\setminus \{u\}\) and *u* is increased by 1, and the distances between all other vertex pairs remain unchanged. Thus

implying \(\mu _{\alpha } (G)-\mu _{\alpha } (G_{uv})>0\), i.e., \(\mu _{\alpha }(G)>\mu _{\alpha }(G_{uv})\). □

The previous theorem has been established for \(\alpha =0,\frac{1}{2}\) in [16, 25].

### Theorem 4.2

*Suppose that**G**is a connected graph with**k**edge*-*disjoint nontrivial induced subgraphs*\(G_{1}, \ldots , G_{k}\)*such that*\(V(G_{i})\cap V(G_{j})=\{u\}\)*for*\(1\le i< j\le k\)*and*\(\bigcup_{i=1}^{k}V(G_{i})=V(G)\), *where*\(k\ge 3\). *Let*\(\emptyset \ne K\subseteq \{3, \ldots , k\}\)*and let*\(N_{K}=\bigcup_{i\in K}N_{G_{i}}(u)\). *For*\(v'\in V(G_{1})\setminus \{u\}\)*and*\(v''\in V(G_{2})\setminus \{u\}\), *let*

*and*

*Then*\(\mu _{\alpha }(G)< \max \{\mu _{\alpha }(G'), \mu _{\alpha }(G'')\}\).

### Proof

Let *x* be the distance *α*-Perron vector of *G*. Let \(V_{K}= (\bigcup_{i\in K} V(G_{i}) )\setminus \{u\}\). Let

As we pass from *G* to \(G'\), the distance between a vertex in \(V(G_{2})\) and a vertex in \(V_{K}\) is increased by \(d_{G}(u,v')\), the distance between a vertex *w* in \(V(G_{1})\setminus \{u\}\) and a vertex in \(V_{K}\) is decreased by \(d_{G}(w,u)-d_{G}(w,v')\), which is at most \(d_{G}(u,v')\), and the distances between all other vertex pairs are increased or remain unchanged. Thus

If \(\varGamma \ge 0\), then \(\mu _{\alpha } (G')-\mu _{\alpha } (G)> d_{G}(u,v')\varGamma \ge 0\), implying \(\mu _{\alpha } (G)<\mu _{\alpha } (G')\). Suppose that \(\varGamma <0\). As we pass from *G* to \(G''\), the distance between a vertex in \(V(G_{1})\) and a vertex in \(V_{K}\) is increased by \(d_{G}(u,v'')\), the distance between a vertex *w* in \(V(G_{2})\setminus \{u\}\) and a vertex in \(V_{K}\) is decreased by \(d_{G}(w,u)-d_{G}(w,v'')\), which is at most \(d_{G}(u,v'')\), and the distances between all other vertex pairs are increased or remain unchanged. Thus

implying \(\mu _{\alpha } (G'')-\mu _{\alpha } (G)> 0\), i.e., \(\mu _{\alpha }(G)<\mu _{\alpha }(G'')\). □

Weak versions of previous theorem for \(\alpha =0\) have been given in [28, 30] and a weak version for \(\alpha =\frac{1}{2}\) may be found in [16].

For positive integer *p* and a graph *G* with \(u\in V(G)\), let \(G(u;p)\) be the graph obtained from *G* by attaching a pendant path of length *p* at *u*. Let \(G(u;0)=G\), and in this case a pendant path of length 0 is understood the trivial path consisting of a single vertex *u*.

For nonnegative integers *p*, *q* and a graph *G*, let \(G_{u}(p,q)\) be the graph \(H(u;q)\) with \(H=G(u;p)\). The following corollary has been known for \(\alpha =0\) in [24, 28] and \(\alpha =\frac{1}{2}\) in [15, 16].

### Corollary 4.1

*Let**H**be a nontrivial connected graph with*\(u\in V(H)\). *If*\(p\ge q\ge 1\), *then*\(\mu _{\alpha } (H_{u}(p,q))<\mu _{\alpha } (H_{u}(p+1,q-1))\).

### Proof

Let \(G=H_{u}(p,q)\). Let \(P=uu_{1}\cdots u_{p}\) and \(Q=uv_{1}\cdots v_{q}\) be two pendant paths of lengths *p* and *q*, respectively, in *G*. Using the notations in Theorem 4.2 with \(k=3\), \(G_{1}=P\), \(G_{2}=Q\), \(G_{3}=H\), \(v'=u_{p-q+1}\) and \(v''=v_{1}\), we have \(G'\cong G'' \cong H_{u}(p+1,q-1)\), and thus by Theorem 4.2, we have \(\mu _{\alpha } (H_{u}(p,q))<\mu _{\alpha } (H_{u}(p+1,q-1))\). □

### Theorem 4.3

*Suppose that**G**is a connected graph with three edge*-*disjoint induced subgraphs*\(G_{1}\), \(G_{2}\)*and*\(G_{3}\)*such that*\(V(G_{1})\cap V(G_{3})=\{u\}\), \(V(G_{2})\cap V(G_{3})=\{v\}\), \(\bigcup_{i=1}^{3}V(G_{i})=V(G)\), *and*\(G_{1}-u\), \(G_{2}-v\), *and*\(G_{3}-u-v\)*are all nontrivial*. *Suppose that*\(uv\in E(G_{3})\). *For*\(u'\in N_{G_{1}}(u) \)*and*\(v'\in N_{G_{2}}(v)\), *let*

*and*

*where*\(H=G-\{uw: w\in N_{G_{3}-uv}(u)\}-\{vw: w\in N_{G_{3}-uv}(v)\}\). *Then*\(\mu _{\alpha } (G)<\mu _{\alpha } (G')\)*or*\(\mu _{\alpha } (G)<\mu _{\alpha } (G'')\).

### Proof

Let *x* be the distance *α*-Perron vector of *G*. Let

As we pass from *G* to \(G'\), the distance between a vertex in \(V(G_{2})\) and a vertex in \(V(G_{3})\setminus \{u,v\}\) is increased by 1, the distance between a vertex in \(V(G_{1})\) and a vertex in \(V(G_{3})\setminus \{u,v\}\) may be increased, unchanged, or decreased by 1, and the distances between any other vertex pairs remain unchanged. Thus

If \(\varGamma \ge 0\), then \(\mu _{\alpha } (G')-\mu _{\alpha } (G)\ge 0\), i.e., \(\mu _{\alpha } (G)\le \mu _{\alpha } (G')\). If \(\mu _{\alpha } (G)=\mu _{\alpha } (G')\), then \(\mu _{\alpha } (G')=x^{\top }D_{\alpha }(G')x\), implying *x* is the distance *α*-Perron vector of \(G'\). By the *α*-equations of *G* and \(G'\) at *v*, we have

a contradiction. Thus, if \(\varGamma \ge 0\), then \(\mu _{\alpha } (G)<\mu _{\alpha } (G')\).

Suppose that \(\varGamma <0\). As earlier, we have

and thus \(\mu _{\alpha }(G)<\mu _{\alpha }(G'')\). □

A weak version of previous theorem for \(\alpha =\frac{1}{2}\) has been established in [16].

For nonnegative integers *p*, *q* and a graph *G* with \(u,v\in V(G)\), let \(G_{u,v}(p,q)\) be the graph \(H(v;q)\) with \(H=G(u;p)\). The following corollary has been known for \(\alpha =0,\frac{1}{2}\) in [15, 32].

### Corollary 4.2

*Let**H**be a connected graph of order at least* 3 *with*\(uv\in E(H)\). *Suppose that*\(\eta (u)=v\)*for some automorphism**η**of**G*. *For*\(p\ge q\ge 1\), *we have*\(\mu _{\alpha }(H_{u,v}(p,q))<\mu _{\alpha }(H_{u,v}(p+1,q-1))\).

### Proof

Let \(G=H_{u,v}(p,q)\). Let \(P=uu_{1}\cdots u_{p}\) and \(Q=vv_{1}\cdots v_{q}\) be two pendant paths of lengths *p* and *q* in *G* at *u* and *v*, respectively. Using the notations of Theorem 4.3 with \(G_{1}=P\), \(G_{2}=Q\), \(G_{3}=H\), \(u'=u_{1}\) and \(v'=v_{1}\), we have \(G'\cong H_{u,v}(p-1,q+1)\) and \(G''\cong H_{u,v}(p+1,q-1)\), and thus by Theorem 4.3, we have \(\mu _{\alpha } (H_{u,v}(p,q))<\max \{\mu _{\alpha }(H_{u,v}(p-1,q+1)), \mu _{\alpha }(H_{u,v}(p+1,q-1))\}\). If \(p=q\) (\(p=q+1\), respectively), then \(H_{u,v}(p-1,q+1)\cong H_{u,v}(p+1,q-1)\) (\(H_{u,v}(p,q)\cong H_{u,v}(p-1,q+1)\), respectively) as \(\eta (u)=v\) for some automorphism *η* of *G*, and thus from the above inequality, we have \(\mu _{\alpha } (G)<\mu _{\alpha }(H_{u,v}(p+1,q-1))\). Suppose that \(p\ge q+2\) and \(\mu _{\alpha } (G)<\mu _{\alpha }(H_{u,v}(p-1,q+1))\). If \(p\not \equiv q \pmod{2}\), then we have

which is impossible. If \(p\equiv q \pmod{2}\), then we have

which is also impossible. Therefore \(\mu _{\alpha } (H_{u,v}(p,q))<\mu _{\alpha } (H_{u,v}(p+1,q-1))\). □

## 5 Graphs with small or large distance *α*-spectral radius

First we determine the graphs with minimum distance *α*-spectral radius among trees and unicyclic graphs.

### Theorem 5.1

*Let**G**be a tree of order**n*. *Then*\(\mu _{\alpha }(G)\geq \mu _{\alpha }(S_{n})\)*with equality if and only if*\(G\cong S_{n}\).

### Proof

The result is trivial if \(n=1,2,3\). Suppose that \(n\ge 4\). Let *G* be a tree of order *n* such that \(\mu _{\alpha }(G)\) is as small as possible. Let *d* be the diameter of *G*. Evidently, \(d\geq 2\). Suppose that \(d\ge 3\). Let \(v_{0}v_{1}\cdots v_{d}\) be a diametral path of *G*. By Theorem 4.1, \(\mu _{\alpha }(G_{v_{1}v_{2}})<\mu _{\alpha }(G)\), a contradiction. Thus \(d= 2\), i.e., \(G\cong S_{n}\). □

In Theorem 5.1, the case \(\alpha =0\) has been known in [24] and the case \(\alpha =\frac{1}{2}\) has been known in [16, 29].

For \(n-1\ge 3\) and \(1\le a\le \lfloor \frac{n-2}{2} \rfloor \), let \(D_{n,a}\) be the tree obtained from vertex-disjoint \(S_{a+1}\) with center *u* and \(S_{n-a-1}\) with center *v* by adding an edge *uv*. Let *T* be a tree of order *n* with minimum distance *α*-spectral radius, where \(T\ncong S_{n}\). Let *d* be the diameter of *T*. Then \(d\geq 3\). Suppose that \(d\geq 4\). Let \(v_{0}v_{1}\cdots v_{d}\) be a diametral path of *T*. Note that \(T_{v_{1}v_{2}} \ncong S_{n}\). By Theorem 4.1, \(\mu _{\alpha }(T_{v_{1}v_{2}})<\mu _{\alpha }(T)\), a contradiction. Thus \(d=3\), implying \(T\cong D_{n,a}\) for some *a* with \(1\leq a\leq \lfloor \frac{n-2}{2}\rfloor \).

Let \(S_{n}^{+}\) is the graph obtained from \(S_{n}\) by adding an edge between two vertices of degree one.

### Lemma 5.1

([29])

*Let**G**be a unicyclic graph of order*\(n\ge 6\). *If*\(G\ncong S_{n}^{+}\), *then*

Note that for \(n=5\), we have \(\sigma (C_{n})=\sigma (S_{n}^{+})\). So, in the above lemma, the condition \(n\ge 6\) is necessary.

### Theorem 5.2

*Let**G**be a unicyclic graph of order*\(n\ge 8\). *Then*\(\mu _{\alpha }(G)\ge \mu _{\alpha }(S_{n}^{+})\)*with equality if and only if*\(G\cong S_{n}^{+}\).

### Proof

Suppose that \(G\ncong S_{n}^{+}\). We only need to show that \(\mu _{\alpha }(G)>\mu _{\alpha }(S_{n}^{+})\).

By Lemmas 2.4 and 5.1, we have

By [20, p. 24, Theorem 1.1] or by Theorem 3.2, we have

Since \(n\ge 8\), we have

as desired. □

The result in Theorem 5.2 for \(\alpha =0, \frac{1}{2}\) has been known in [29, 31].

In the following, we determine the graphs with maximum distance *α*-spectral radius among some classes of graphs.

For \(2\le \Delta \le n-1\), let \(B_{n,\Delta }\) be a tree obtained by attaching \(\Delta -1\) pendant vertices to a terminal vertex of the path \(P_{n-\Delta +1}\). In particular, \(B_{n,2}=P_{n}\) and \(B_{n, n-1}=S_{n}\). The following theorem for \(\alpha =0,\frac{1}{2}\) was given in [16, 24] for trees.

### Theorem 5.3

*Let**G**be a connected graph of order**n**with maximum degree* Δ, *where*\(2\le \Delta \le n-1\). *Then*\(\mu _{\alpha } (G)\leq \mu _{\alpha } (B_{n,\Delta })\)*with equality if and only if*\(G\cong B_{n,\Delta }\).

### Proof

Let *G* be a graph among connected graphs of order *n* with maximum degree Δ such that \(\mu _{\alpha } (G)\) is as large as possible. Then *G* has a spanning tree *T* with maximum degree Δ. By Lemma 2.3, \(\mu _{\alpha }(G)\le \mu _{\alpha }(T)\) with equality if and only if \(G\cong T\). Thus *G* is a tree.

The result is trivial if \(n=3,4\) and if \(\Delta =2, n-1\). Suppose that \(3\leq \Delta \leq n-2\). We only need to show that \(G\cong B_{n,\Delta }\).

Let \(u\in V(G)\) with \(\operatorname{deg}_{G}(u)=\Delta \). Suppose that there exists a vertex different from *u* with degree at least 3. Then we may choose such a vertex *w* of degree at least 3 such that \(d_{G}(u,w)\) is as large as possible. Obviously, there are two pendant paths, say *P* and *Q*, at *w* of lengths at least 1. Let *p* and *q* be the lengths of *P* and *Q*, respectively. Assume that \(p\ge q\). Let \(H=G[V(G)\setminus ((V(P)\cup V(Q))\setminus \{w\})]\). Then \(G\cong H_{w}(p,q)\). Note that \(G'=H_{w}(p+1,q-1)\) is a tree of order *n* with maximum degree Δ. By Corollary 4.1, \(\mu _{\alpha }(G)<\mu _{\alpha }(G')\), a contradiction. Then *u* is the unique vertex of *G* with degree at least 3, and thus *G* consists of Δ pendant paths, say \(Q_{1}, \ldots , Q_{\Delta }\) at *u*. If two of them, say \(Q_{i}\) and \(Q_{j}\) with \(i\ne j\) are of lengths at least 2, then \(G\cong H'_{u}(r,s)\), where \(H'=G[V(G)\setminus ((V(Q_{i})\cup V(Q_{j}))\setminus \{u\})]\), and *r* and *s* are the lengths of \(Q_{i}\) and \(Q_{j}\), respectively. Assume that \(r\ge s\). Obviously, \(G''=H'_{u}(r+1,s-1)\) is a tree of order *n* with maximum degree Δ. By Corollary 4.1, \(\mu _{\alpha } (G)< \mu _{\alpha } (G'')\), also a contradiction. Thus there is exactly one pendant path at *u* of length at least 2, implying \(G\cong B_{n,\Delta }\). □

If *G* is a connected graph of order 1 or 2, then \(G\cong P_{n}\). If *G* is a connected graph of order 3, then \(G\cong P_{3}\), \(K_{3}\), and by Lemma 2.3, \(\mu _{\alpha }(K_{3})<\mu _{\alpha }(P_{3})\).

Ruzieh and Powers [23] showed that \(P_{n}\) is the unique connected graph of order *n* with maximum distance 0-spectral radius, and it was proved in [25] that \(B_{n,3}\) is the unique tree of order *n* different from \(P_{n}\) with maximum distance 0-spectral radius. For \(\alpha =\frac{1}{2}\), the following theorem was given in [16].

### Theorem 5.4

*Let**G**be a connected graph of order*\(n\geq 4\), *where*\(G\ncong P_{n}\). *Then*\(\mu _{\alpha }(G)\leq \mu _{\alpha }(B_{n,3})<\mu _{\alpha }(P_{n})\)*with equality if and only if*\(G\cong B_{n,3}\).

### Proof

First suppose that *G* is a tree. If \(n=4\), then the result follows from Theorem 4.1. Suppose that \(n\geq 5\). Let Δ be the maximum degree of *G*. Since \(G\ncong P_{n}\), we have \(\Delta \geq 3\). By Theorem 5.3, \(\mu _{\alpha }(G)\le \mu _{\alpha }(B_{n,\Delta })\) with equality if and only if \(G\cong B_{n, \Delta }\). By Corollary 4.1, \(\mu _{\alpha }(G)\le \mu _{\alpha }(B_{n,\Delta })\le \mu _{\alpha }(B_{n,3})< \mu _{\alpha }(P_{n})\) with equalities if and only if \(\Delta =3\) and \(G\cong B_{n, \Delta }\), i.e., \(G\cong B_{n,3}\).

Now suppose that *G* is not a tree. Then *G* contains at least one cycle. If there is a spanning tree *T* with \(T\ncong P_{n}\), then by Lemma 2.3 and the above argument, we have \(\mu _{\alpha }(G)< \mu _{\alpha }(T)\leq \mu _{\alpha }(B_{n,3})\). If any spanning tree of *G* is a path, then *G* is a cycle \(C_{n}\). Now we only need to show that \(\mu _{\alpha }(C_{n})<\mu _{\alpha }(B_{n,3})\).

Let \(C_{n}=u_{1}u_{2}\cdots u_{n}u_{1}\) and \(T'=C_{n}-\{u_{1}u_{2}, u_{2}u_{3}\}+u_{2}u_{n}\). Then \(T'\cong B_{n,3}\). Let *x* be the distance *α*-Perron vector of \(C_{n}\). By Lemma 2.3, we have \(x_{u_{1}}=\cdots =x_{u_{n}}\). As we pass from \(C_{n}\) to \(T'\), the distance between \(u_{2}\) and \(u_{1}\) is increased by 1, the distance between \(u_{2}\) and \(u_{i}\) with \(3\leq i\leq \lceil \frac{n+1}{2} \rceil \) is increased by \(n-2i+3\), the distance between \(u_{2}\) and \(u_{i}\) with \(\lfloor \frac{n+1}{2} \rfloor +2\leq i\leq n\) is decreased by 1, and the distances between all other vertex pairs are increased or remain unchanged. Thus

and therefore \(\mu _{\alpha }(C_{n})<\mu _{\alpha }(B_{n,3})\), as desired. □

A clique of *G* is a subset of vertices whose induced subgraph is a complete graph, and the clique number of *G* is the maximum number of vertices in a clique of *G*. For \(2\le \omega \le n\). Let \(Ki_{n,\omega }\) be the graph obtained from a complete graph \(K_{\omega }\) and a path \(P_{n-\omega }\) by adding an edge between a vertex of \(K_{\omega }\) and a terminal vertex of \(P_{n-\omega }\) if \(\omega < n\) and let \(Ki_{n,\omega }=K_{n}\) if \(\omega =n\). In particular, \(Ki_{n,2}\cong P_{n}\) for \(n\ge 2\). The following result for \(\alpha =0,\frac{1}{2}\) was given in [15, 21].

### Theorem 5.5

*Let**G**be a connected graph of order*\(n\geq 2\)*with clique number*\(\omega \geq 2\). *Then*\(\mu _{\alpha } (G)\leq \mu _{\alpha } (Ki_{n,\omega })\)*with equality if and only if*\(G\cong Ki_{n,\omega }\).

### Proof

It is trivial if \(\omega =n\) and it follows from Theorem 5.4 if \(\omega =2\).

Suppose that \(3\le \omega \le n-1\). Let *G* be a graph among connected graphs of order *n* with clique number *ω* such that \(\mu _{\alpha } (G)\) is as large as possible. We only need to show that \(G\cong Ki_{n,\omega }\).

Let \(S=\{v_{1},\ldots ,v_{\omega }\}\) be a clique of *G*. By Lemma 2.3, \(G-E(G[S])\) is a forest. Let \(T_{i}\) be the component of \(G-E(G[S])\) containing \(v_{i}\), where \(1\leq i\leq \omega \). For \(1\leq i\leq \omega \), by Corollary 4.1, if \(T_{i}\) is nontrivial, then \(T_{i}\) is a pendant path at \(v_{i}\). Note that any two distinct vertices in \(G[S]\) are adjacent. By Corollary 4.2, there is only one nontrivial \(T_{i}\), and thus \(G\cong Ki_{n,\omega }\). □

Recall that \(Ki_{n,3}\) is the unique unicyclic graph of order \(n\ge 3\) with maximum distance 0-spectral radius [31], and the unique odd-cycle unicyclic graph of order \(n\ge 3\) with maximum distance \(\frac{1}{2}\)-spectral radius [15].

### Theorem 5.6

*Let**G**be a unicyclic odd*-*cycle graph of order*\(n\ge 3\). *Then*\(\mu _{\alpha } (G)\leq \mu (Ki_{n,3})\)*with equality if and only if*\(G\cong Ki_{n,3}\).

### Proof

If \(n=3,4\), the result is trivial. Suppose that \(n\geq 5\). Let *G* be a graph with maximum distance *α*-spectral radius among unicyclic odd-cycle graphs of order *n*. We only need to show that \(G\cong Ki_{n,3}\).

Let \(C=v_{1} \cdots v_{2k+1}v_{1}\) be the unique cycle of *G*, where \(k\ge 1\). Let \(T_{i}\) be the component of \(G-E(C)\) containing \(v_{i}\) for \(1\le i\le 2k+1\). Let \(U_{1}= V(T_{2k})\cup V(T_{2k+1})\), \(U_{2}=\bigcup_{k+1\le i\le 2k-1} V(T_{i})\) and \(U_{3}=\bigcup_{1\le i\le k-1} V(T_{i})\). Let *x* be the distance *α*-Perron vector of *G*. Let

Suppose that \(k\ge 2\). Let \(G'=G-v_{1}v_{2k+1}+v_{2k+1}v_{2k-1}\). Note that the length of *C* is odd. As we pass from *G* to \(G'\), the distance between a vertex in \(S_{1}\) and a vertex in \(S_{3}\) is increased by at least 1, the distance between \(S_{2}\) and \(V(T_{2k+1})\) is decreased by 1, and the distance between all other vertex pairs are increased or remain unchanged. Thus

If \(\varGamma \ge 0\), then \(\mu _{\alpha }(G')>\mu _{\alpha }(G)\), a contradiction. Thus \(\varGamma <0\). Let \(G''=G-v_{2k}v_{2k-1}+v_{2k}v_{1}\). As we pass from *G* to \(G''\), the distance between a vertex in \(S_{1}\) and a vertex in \(U_{2}\) is increased by at least 1, the distance between \(U_{3}\) and \(V(T_{2k})\) is decreased by 1, and the distance between all other vertex pairs are increased or remain unchanged. As above, we have

Thus \(\mu _{\alpha }(G'')>\mu _{\alpha }(G)\), also a contradiction. It follows that \(k=1\), i.e., the unique cycle of *G* is of length 3.

Obviously, \(T_{i}\) is a tree for \(1\le i\le 3\). For \(1\le i\le 3\), by Corollary 4.1, if \(T_{i}\) is nontrivial, then it is a path with a terminal vertex \(v_{i}\). Then by Corollary 4.2, only one \(T_{i}\) is nontrivial. Thus \(G\cong Ki_{n,3}\). □

Let *G* be a unicyclic graph of order \(n\ge 4\) with maximum distance *α*-spectral radius. By Corollary 4.1, the maximum degree of *G* is 3 and all vertices of degree 3 lie on the unique cycle. Let *u* be a vertex of degree 3 and *P* be the pendant path at *u*. Let *v* and *w* be the two neighbors of *u* on the cycle, and *z* the neighbor of *u* on *P*. Let \(G_{1}=G-uw+vw\) and \(G_{2}=G-uw+wz\). Then \(\mu _{\alpha }(G)<\max \{\mu _{\alpha }(G_{1}), \mu _{\alpha }(G_{2}) \}\) if the length of the cycle of *G* is odd, see [4, Lemma 6.11]. Note that the argument does not work when the length of the cycle of *G* is even. So we need other ways to determine the unicyclic graph(s) with maximum distance *α*-spectral radius even for \(\alpha =\frac{1}{2}\).

## 6 Remarks

In this paper, we study the distance *α*-spectral radius of a connected graph. We consider bounds for the distance *α*-spectral radius, local transformations to change the distance *α*-spectral radius, and the characterizations for graphs with minimum and/or maximum distance *α*-spectral radius in some classes of connected graphs.

Besides the distance *α*-spectral radius, we may concern other eigenvalues of \(D_{\alpha }(G)\) for a connected graph *G*. We give examples.

For an \(n\times n\) Hermitian matrix *C*, let \(\lambda _{1}(C), \ldots , \lambda _{n}(C)\) be the eigenvalues of *C*, arranged in a nonincreasing order. Let *A*, *B* be \(n\times n\) Hermitian matrices. Weyl’s inequalities [13, p. 181] state that

and

Using these inequalities, and as in the recent work of Atik and Panigrahi [3], we have

### Theorem 6.1

*Let**G**be a connected graph and**λ**be any eigenvalue of*\(D_{\alpha }(G)\)*other than the distance**α*-*spectral radius*. *Then*

### Proof

Let \(D_{\alpha }(G)=A+B\), where \(A=(\alpha T_{\min }(G)-(1-\alpha ))I_{n}+(1-\alpha )J_{n\times n}\). Then *B* is a nonnegative symmetric matrix with maximum row sum \(T_{\max }(G)-\alpha T_{\min }(G)-(1-\alpha )(n-1)\). Thus \(|\lambda _{n}(B)|\le \lambda _{1}(B)\le T_{\max }(G)-\alpha T_{\min }(G)-(1- \alpha )(n-1)\).

For matrix *A*, we have \(\lambda _{1}(A)=\alpha T_{\min }(G)+(1-\alpha )(n-1)\) and \(\lambda _{j}(A)=\alpha T_{\min }(G)-1+\alpha \) for \(j=2,\ldots ,n\). Thus, for \(j=2,\ldots , n\), we have by the above Weyl’s inequalities that

and

This completes the proof. □

Let *G* be a connected graph and *λ* be any eigenvalue of \(D_{\alpha }(G)\) other than the distance *α*-spectral radius. By previous theorem, we have

The distance *α*-energy of a connected graph *G* of order *n* is defined as

Then \(\mathcal{E}_{0}(G)\) is the distance energy of *G* [14, 33], while

is half of the distance signless Laplacian energy of *G* [8]. Thus, it is possible to study the distance energy and the distance signless Laplacian energy in a unified way.

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Guo, H., Zhou, B. On the distance *α*-spectral radius of a connected graph.
*J Inequal Appl* **2020**, 161 (2020). https://doi.org/10.1186/s13660-020-02427-4

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DOI: https://doi.org/10.1186/s13660-020-02427-4