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On the distance α-spectral radius of a connected graph

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.

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

$$ D_{\alpha }(G)=\alpha T(G)+(1-\alpha )D(G), $$

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.

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

$$ x^{\top }D_{\alpha }(G)x=\alpha \sum _{u\in V(G)}T_{G}(u)x_{u}^{2}+2 \sum_{\{u,v\}\subseteq V(G)}(1-\alpha )d_{G}(u,v)x_{u}x_{v}, $$

or equivalently,

$$ x^{\top }D_{\alpha }(G)x=\sum_{\{u,v\}\subseteq V(G)}d_{G}(u,v) \bigl( \alpha \bigl(x_{u}^{2}+x_{v}^{2} \bigr)+2(1-\alpha )x_{u}x_{v} \bigr). $$

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 \(u\in V(G)\),

$$ \mu _{\alpha }(G)x_{u}=\sum_{v\in V(G)}d_{G}(u,v) \bigl(\alpha x_{u}+(1- \alpha )x_{v}\bigr), $$

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 thatGis a connected graph, ηis an automorphism ofG, andxis the distanceα-Perron vector ofG. 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 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 thatAandBare square nonnegative matrices, Ais irreducible, and\(A-B\)is nonnegative but nonzero. Then\(\rho (A)> \rho (B)\).

By Lemma 2.2, we have

Lemma 2.3

Suppose thatGis a connected graph with\(u,v\in V(G)\), anduandvare 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 thatGis a connected graph of ordern. Then\(\mu _{\alpha }(G)\ge \frac{2\sigma (G)}{n}\)with equality if and only ifGis transmission regular.

For an \(n\times n\) nonnegative matrix \(A=(a_{ij})\), let \(r_{i}\) be the ith 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 somepandqwith\(1 \le p\), \(q\le n \), \(\ell =\max \{r_{i}-a_{ip}: i\in S\setminus \{p\}\}\), \(m=\min \{r_{i}-a_{iq}: i\in S\setminus \{q\}\}\), \(s=\max \{a_{ip}: i\in S\setminus \{p\}\}\)and\(t=\min \{a_{iq}: i\in S\setminus \{q\}\}\). Then

$$\begin{aligned}& \frac{a_{qq}+m+\sqrt{(m-a_{qq})^{2}+4t(r_{\max }-a_{qq})}}{2} \\& \quad \le \rho (A) \\& \quad \le \frac{a_{pp}+\ell +\sqrt{(\ell -a_{pp})^{2}+4s(r_{\min }-a_{pp})}}{2}. \end{aligned}$$

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.

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

$$\begin{aligned} \mu _{\alpha }(G) \ge & \frac{1}{2} \bigl( \alpha \bigl(4n-4- \Delta -\Delta '\bigr) \\ &{} +\sqrt{\alpha ^{2}\bigl(4n-4-\Delta -\Delta ' \bigr)^{2}-4(2\alpha -1) (2n-2- \Delta ) \bigl(2n-2-\Delta '\bigr)} \bigr) \end{aligned}$$

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

$$\begin{aligned} \mu _{\alpha }(G) \le & \frac{1}{2}\large \bigl( \alpha \bigl(2dn-2-(d-1) \bigl(d+\delta +\delta '\bigr)\bigr) \\ &{}+\sqrt{\alpha ^{2}\bigl(2dn-2-(d-1) \bigl(d+\delta +\delta '\bigr)\bigr)^{2}-4(2 \alpha -1) S S'} \bigr) \end{aligned}$$

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

LetGbe a connected graph anduandvbe 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

$$\begin{aligned}& \frac{m_{2}+\alpha T_{\max }(G)+\sqrt{(m_{2}-\alpha T_{\max }(G))^{2}+4(1-\alpha )^{2}T_{\max }(G)}}{2} \\& \quad \le \mu _{\alpha }(G) \\& \quad \le \frac{m_{1}+\alpha T_{\min }(G)+\sqrt{(m_{1}-\alpha T_{\min }(G))^{2}+4(1-\alpha )^{2}e(u)T_{\min }(G)}}{2}. \end{aligned}$$

The first equality holds if and only ifGis a complete graph and the second equality holds if and only ifGis 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

$$ \mu _{0}(G)< \varLambda - \frac{n\varLambda -2\sigma (G)}{(n\varLambda -2\sigma (G)+1)n} $$

and

$$ \mu _{1/2}(G)< \varLambda - \frac{n\varLambda -2\sigma (G)}{(2(n\varLambda -2\sigma (G))+1)n}. $$

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

$$ \mu _{0}(G)< \varLambda - \frac{n\varLambda -2\sigma (G)}{(n\varLambda -2\sigma (G))\frac{4\sigma (G)}{n\varLambda }+n} $$

and

$$ \mu _{1/2}(G)< \varLambda - \frac{n\varLambda -2\sigma (G)}{(n\varLambda -2\sigma (G))\frac{8\sigma (G)}{n\varLambda }+n}. $$

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

Theorem 3.2

LetGbe a connected non-transmission-regular graph of ordern. Then

$$ \mu _{\alpha }(G)< \varLambda - \frac{(1-\alpha )n\varLambda (n\varLambda -2\sigma (G))}{4\sigma (G)(n\varLambda -2\sigma (G))+(1-\alpha )n^{2}\varLambda }, $$

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

$$\begin{aligned} \mu _{\alpha }(G) =&x^{\top }D_{\alpha }(G)x \\ =&\alpha \sum_{w\in V(G)}T_{G}(w)x_{w}^{2}+2(1- \alpha )\sum_{\{w,z \}\subseteq V(G)}d_{wz}x_{w}x_{z} \\ < & 2\alpha \sigma (G)x_{u}^{2}+2(1-\alpha )\sigma (G)x_{u}^{2}, \end{aligned}$$

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

$$\begin{aligned}& \varLambda -\mu _{\alpha }(G) \\& \quad = \varLambda -\alpha \sum_{w\in V(G)}T_{G}(w)x_{w}^{2}-2(1- \alpha ) \sum_{\{w,z\}\subseteq V(G)}d_{wz}x_{w}x_{z} \\& \quad = \sum_{w\in V(G)}\bigl(\varLambda -T_{G}(w) \bigr)x_{w}^{2}+(1-\alpha )\sum _{ \{w,z\}\subseteq V(G)}d_{wz}(x_{w}-x_{z})^{2} \\& \quad \ge \sum_{w\in V(G)}\bigl(\varLambda -T_{G}(w)\bigr)x_{v}^{2}+(1-\alpha )\sum _{ \{w,z\}\subseteq V(G)}d_{wz}(x_{w}-x_{z})^{2} \\& \quad = \bigl(n\varLambda -2\sigma (G)\bigr)x_{v}^{2}+(1- \alpha )\sum_{\{w,z\} \subseteq V(G)}d_{wz}(x_{w}-x_{z})^{2}. \end{aligned}$$

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,

$$ \sum_{\{w,z\}\subseteq V(G)}d_{wz}(x_{w}-x_{z})^{2} \ge N_{1}+N_{2}, $$

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

$$ d_{wu}(x_{w}-x_{u})^{2}+d_{wv}(x_{w}-x_{v})^{2} \ge (x_{w}-x_{u})^{2}+(x_{w}-x_{v})^{2} \ge \frac{1}{2}(x_{u}-x_{v})^{2}, $$

and thus

$$\begin{aligned} N_{1} \ge & \sum_{w\in V(G)\setminus V(P)} \bigl(d_{wu}(x_{w}-x_{u})^{2}+d_{wv}(x_{w}-x_{v})^{2} \bigr) \\ \ge & \sum_{w\in V(G)\setminus V(P)} \frac{1}{2}(x_{u}-x_{v})^{2} \\ = &\frac{n-\ell -1}{2}(x_{u}-x_{v})^{2}. \end{aligned}$$

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

$$\begin{aligned}& d_{w_{0}w_{i}}(x_{w_{0}}-x_{w_{i}})^{2}+d_{w_{i} w_{\ell }}(x_{w_{i}}-x_{w_{\ell }})^{2} \\& \quad \ge \min \{i,\ell -i\} \bigl((x_{w_{0}}-x_{w_{i}})^{2}+(x_{w_{i}}-x_{w_{\ell }})^{2} \bigr) \\& \quad \ge \min \{i,\ell -i\}\cdot \frac{1}{2}(x_{w_{0}}-x_{w_{\ell }})^{2} \\& \quad = \frac{1}{2}\min \{i,\ell -i\}(x_{u}-x_{v})^{2}, \end{aligned}$$

and thus

$$\begin{aligned} N_{2} \ge & d_{uv}(x_{u}-x_{v})^{2}+ \sum_{i=1}^{\ell -1} \bigl(d_{w_{0}w_{i}}(x_{w_{i}}-x_{w_{0}})^{2}+d_{w_{i} w_{\ell }}(x_{w_{i}}-x_{w_{\ell }})^{2} \bigr) \\ \ge & \ell (x_{u}-x_{v})^{2}+\sum _{i=1}^{\ell -1}\frac{1}{2}\min \{i, \ell -i\}(x_{u}-x_{v})^{2} \\ =& \Biggl(\ell +\frac{1}{2}\sum_{i=1}^{\ell -1} \min \{i,\ell -i\} \Biggr) (x_{u}-x_{v})^{2} \\ =& \textstyle\begin{cases} \frac{\ell ^{2}+8\ell }{8}(x_{u}-x_{v})^{2} & \text{if $\ell $ is even}, \\ \frac{\ell ^{2}+8\ell -1}{8}(x_{u}-x_{v})^{2} & \text{if $\ell $ is odd}. \end{cases}\displaystyle \end{aligned}$$

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

In this case, we have

$$\begin{aligned} \sum_{\{w,z\}\subseteq V(G)}d_{wz}(x_{w}-x_{z})^{2} \geq & N_{1}+N_{2} \\ \ge & \frac{n-1-1}{2}(x_{u}-x_{v})^{2}+(x_{u}-x_{v})^{2} \\ =& \frac{n}{2}(x_{u}-x_{v})^{2}. \end{aligned}$$

Thus

$$\begin{aligned} \varLambda -\mu _{\alpha }(G) \ge & \bigl(n\varLambda -2\sigma (G) \bigr)x_{v}^{2}+(1- \alpha )\sum _{\{w,z\}\subseteq V(G)}d_{wz}(x_{w}-x_{z})^{2} \\ \ge & \bigl(n\varLambda -2\sigma (G)\bigr)x_{v}^{2}+(1- \alpha )\frac{n}{2}(x_{u}-x_{v})^{2}. \end{aligned}$$

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

$$\begin{aligned} \varLambda -\mu _{\alpha }(G) >&\frac{(1-\alpha )n(n\varLambda -2\sigma (G))}{2(n\varLambda -2\sigma (G))+(1-\alpha )n} \cdot \frac{\mu _{\alpha }(G)}{2\sigma (G)} \\ =&\frac{(1-\alpha )n(n\varLambda -2\sigma (G))\varLambda }{2\sigma (G)(2(n\varLambda -2\sigma (G))+(1-\alpha )n)} \\ &{}- \frac{(1-\alpha )n(n\varLambda -2\sigma (G))(\varLambda -\mu _{\alpha }(G))}{2\sigma (G)(2(n\varLambda -2\sigma (G))+(1-\alpha )n)}, \end{aligned}$$

which implies that

$$ \varLambda -\mu _{\alpha }(G)> \frac{(1-\alpha )n\varLambda (n\varLambda -2\sigma (G))}{4\sigma (G)(n\varLambda -2\sigma (G))+(1-\alpha )n^{2}\varLambda }. $$

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

Suppose first that is even. Then

$$\begin{aligned} \sum_{\{w,z\}\subseteq V(G)}d_{wz}(x_{w}-x_{z})^{2} \geq &N_{1}+N_{2} \\ \ge & \frac{n-\ell -1}{2}(x_{u}-x_{v})^{2}+ \frac{\ell ^{2}+8\ell }{8}(x_{u}-x_{v})^{2} \\ =& \frac{\ell ^{2}+4\ell +4n-4}{8}(x_{u}-x_{v})^{2}. \end{aligned}$$

Thus

$$\begin{aligned} \varLambda -\mu _{\alpha }(G) \ge & \bigl(n\varLambda -2\sigma (G) \bigr)x_{v}^{2}+(1- \alpha )\sum _{\{w,z\}\subseteq V(G)}d_{wz}(x_{w}-x_{z})^{2} \\ \ge & \bigl(n\varLambda -2\sigma (G)\bigr)x_{v}^{2}+(1- \alpha ) \frac{\ell ^{2}+4\ell +4n-4}{8}(x_{u}-x_{v})^{2}. \end{aligned}$$

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

$$ \varLambda -\mu _{\alpha }(G) > \frac{(1-\alpha )(n\varLambda -2\sigma (G))(\ell ^{2}+4\ell +4n-4)}{(1-\alpha )(\ell ^{2}+4\ell +4n-4)+8(n\varLambda -2\sigma (G))} \cdot \frac{\mu _{\alpha }(G)}{2\sigma (G)}, $$

i.e.,

$$ \varLambda -\mu _{\alpha }(G)> \frac{(1-\alpha )(n\varLambda -2\sigma (G))(\ell ^{2}+4\ell +4n-4)\varLambda }{16\sigma (G)(n\varLambda -2\sigma (G))+(1-\alpha )(\ell ^{2}+4\ell +4n-4)n\varLambda }. $$

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

$$\begin{aligned} \varLambda -\mu _{\alpha }(G) >&\frac{(1-\alpha )(n\varLambda -2\sigma (G))(n+2)\varLambda }{4\sigma (G)(n\varLambda -2\sigma (G))+(1-\alpha )(n+2)n\varLambda } \\ >&\frac{(1-\alpha )n\varLambda (n\varLambda -2\sigma (G))}{4\sigma (G)(n\varLambda -2\sigma (G))+(1-\alpha )n^{2}\varLambda }. \end{aligned}$$

Now suppose that is odd. Then

$$\begin{aligned}& \sum_{\{w,z\}\subseteq V(G)}d_{wz}(x_{w}-x_{z})^{2} \\& \quad \geq N_{1}+N_{2} \\& \quad \ge \frac{n-\ell -1}{2}(x_{u}-x_{v})^{2}+ \frac{\ell ^{2}+8\ell -1}{8}(x_{u}-x_{v})^{2} \\& \quad = \frac{\ell ^{2}+4\ell +4n-5}{8}(x_{u}-x_{v})^{2}. \end{aligned}$$

Thus, as early, we have

$$\begin{aligned}& \varLambda -\mu _{\alpha }(G) \\& \quad \ge \bigl(n\varLambda -2\sigma (G)\bigr)x_{v}^{2}+(1- \alpha ) \frac{\ell ^{2}+4\ell +4n-5}{8}(x_{u}-x_{v})^{2} \\& \quad \ge \frac{(1-\alpha )(\ell ^{2}+4\ell +4n-5)(n\varLambda -2\sigma (G))}{8(n\varLambda -2\sigma (G))+(1-\alpha )(\ell ^{2}+4\ell +4n-5)}x_{u}^{2} \\& \quad > \frac{(1-\alpha )(\ell ^{2}+4\ell +4n-5)(n\varLambda -2\sigma (G))}{8(n\varLambda -2\sigma (G))+(1-\alpha )(\ell ^{2}+4\ell +4n-5)} \cdot \frac{\mu _{\alpha }(G)}{2\sigma (G)}, \end{aligned}$$

implying

$$ \varLambda -\mu _{\alpha }(G)> \frac{(1-\alpha )(n\varLambda -2\sigma (G))(\ell ^{2}+4\ell +4n-5)\varLambda }{16\sigma (G)(n\varLambda -2\sigma (G))+(1-\alpha )(\ell ^{2}+4\ell +4n-5)n\varLambda }. $$

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

$$\begin{aligned} \varLambda -\mu _{\alpha }(G) >&\frac{(1-\alpha )(n\varLambda -2\sigma (G))(4+n)\varLambda }{4\sigma (G)(n\varLambda -2\sigma (G))+(1-\alpha )(4+n)n\varLambda } \\ >&\frac{(1-\alpha )n\varLambda (n\varLambda -2\sigma (G))}{4\sigma (G)(n\varLambda -2\sigma (G))+(1-\alpha )n^{2}\varLambda }. \end{aligned}$$

The result follows by combining Cases 1 and 2. □

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 thatGis a connected graph, uvis a cut edge with\(\operatorname{deg}_{G}(u)\ge 2\), andvis adjacent to a pendant vertex\(v'\). Let

$$ G_{uv}=G-\bigl\{ uw: w\in N_{G}(u)\setminus \{v\}\bigr\} +\bigl\{ vw: w\in N_{G}(u) \setminus \{v\}\bigr\} . $$

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

$$\begin{aligned}& \mu _{\alpha } (G)-\mu _{\alpha } (G_{uv}) \\& \quad \geq x^{\top }\bigl(D_{\alpha }(G)-D_{\alpha }(G_{uv}) \bigr)x \\& \quad = \sum_{w\in V(G_{1})\setminus \{u\}}\sum_{z\in V(G_{2}) } \bigl( \alpha \bigl(x_{w}^{2}+x_{z}^{2} \bigr)+2(1-\alpha )x_{w}x_{z} \bigr) \\& \qquad {} -\sum_{ w\in V(G_{1})\setminus \{u\}} \bigl(\alpha \bigl(x_{w}^{2}+x_{u}^{2} \bigr)+2(1-\alpha )x_{w}x_{u} \bigr) \\& \quad \geq \sum_{w\in V(G_{1})\setminus \{u\}} \bigl(\alpha \bigl(x_{w}^{2}+x_{v}^{2} \bigr)+2(1-\alpha )x_{w}x_{v} \bigr) \\& \qquad {} +\sum_{w\in V(G_{1})\setminus \{u\}} \bigl(\alpha \bigl(x_{w}^{2}+x_{v'}^{2} \bigr)+2(1-\alpha )x_{w}x_{v'} \bigr) \\& \qquad {} -\sum_{ w\in V(G_{1})\setminus \{u\}} \bigl(\alpha \bigl(x_{w}^{2}+x_{u}^{2} \bigr)+2(1-\alpha )x_{w}x_{u} \bigr) \\& \quad = \sum_{w\in V(G_{1})\setminus \{u\}} \bigl(\alpha \bigl(x_{w}^{2}+x_{v}^{2} \bigr)+2(1-\alpha )x_{w}x_{v} \bigr) \\& \quad > 0, \end{aligned}$$

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 thatGis a connected graph withkedge-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

$$ G' =G-\{uw: w\in N_{K}\}+\bigl\{ v'w: w\in N_{K}\bigr\} $$

and

$$ G'' =G-\{uw: w\in N_{K}\}+\bigl\{ v''w: w\in N_{K}\bigr\} . $$

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

$$\begin{aligned} \varGamma =&\sum_{w\in V(G_{2})\setminus \{u\}}\sum _{z\in V_{K}} \bigl(\alpha \bigl(x_{w}^{2}+x_{z}^{2} \bigr)+2(1-\alpha )x_{w}x_{z} \bigr) \\ &{}-\sum_{w\in V(G_{1})\setminus \{u\}}\sum_{z\in V_{K}} \bigl( \alpha \bigl(x_{w}^{2}+x_{z}^{2} \bigr)+2(1-\alpha )x_{w}x_{z} \bigr). \end{aligned}$$

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

$$\begin{aligned}& \mu _{\alpha } \bigl(G'\bigr)-\mu _{\alpha } (G) \\& \quad \geq x^{\top }\bigl(D_{\alpha }\bigl(G' \bigr)-D_{\alpha }(G)\bigr)x \\& \quad \geq \sum_{w\in V(G_{2})}\sum _{z\in V_{K}} \bigl(d_{G}\bigl(u,v' \bigr) \bigl(\alpha \bigl(x_{w}^{2}+x_{z}^{2} \bigr)+2(1-\alpha )x_{w}x_{z} \bigr) \bigr) \\& \qquad {} -\sum_{w\in V(G_{1})\setminus \{u\}}\sum_{z\in V_{K}} \bigl(d_{G}\bigl(u,v'\bigr) \bigl(\alpha \bigl(x_{w}^{2}+x_{z}^{2} \bigr)+2(1-\alpha )x_{w}x_{z} \bigr) \bigr) \\& \quad = d_{G}\bigl(u,v'\bigr) \biggl(\varGamma +\sum _{z\in V_{K}} \bigl(\alpha \bigl(x_{u}^{2}+x_{z}^{2} \bigr)+2(1-\alpha )x_{u}x_{z} \bigr) \biggr) \\& \quad > d_{G}\bigl(u,v'\bigr)\varGamma . \end{aligned}$$

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

$$\begin{aligned}& \mu _{\alpha } \bigl(G''\bigr)-\mu _{\alpha } (G) \\& \quad \geq x^{\top }\bigl(D_{\alpha }\bigl(G'' \bigr)-D_{\alpha }(G)\bigr)x \\& \quad \geq \sum_{w\in V(G_{1})}\sum _{z\in V_{K}} \bigl(d_{G}\bigl(u,v'' \bigr) \bigl(\alpha \bigl(x_{w}^{2}+x_{z}^{2} \bigr)+2(1-\alpha )x_{w}x_{z} \bigr) \bigr) \\& \qquad {} -\sum_{w\in V(G_{2})\setminus \{u\}}\sum_{z\in V_{K}} \bigl(d_{G}\bigl(u,v''\bigr) \bigl( \alpha \bigl(x_{w}^{2}+x_{z}^{2} \bigr)+2(1-\alpha )x_{w}x_{z} \bigr) \bigr) \\& \quad = d_{G}\bigl(u,v''\bigr) \biggl(- \varGamma + \sum_{z\in V_{K}} \bigl(\alpha \bigl(x_{u}^{2}+x_{z}^{2} \bigr)+2(1-\alpha )x_{u}x_{z} \bigr) \biggr) \\& \quad > d_{G}\bigl(u,v''\bigr) (-\varGamma ) \\& \quad > 0, \end{aligned}$$

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

LetHbe 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 thatGis 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

$$ G'=H+\bigl\{ u'w: w\in N_{G_{3}-uv}(u)\bigr\} +\bigl\{ uw: w\in N_{G_{3}-uv}(v)\bigr\} $$

and

$$ G''=H+\bigl\{ vw: w\in N_{G_{3}-uv}(u)\bigr\} +\bigl\{ v'w: w\in N_{G_{3}-uv}(v)\bigr\} , $$

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

$$\begin{aligned} \varGamma =&\sum_{w\in V(G_{2})}\sum _{z\in V(G_{3})\setminus \{u,v\} } \bigl(\alpha \bigl(x_{w}^{2}+x_{z}^{2} \bigr)+2(1-\alpha )x_{w}x_{z} \bigr) \\ &{}- \sum_{w\in V(G_{1})}\sum_{z\in V(G_{3})\setminus \{u,v\}} \bigl( \alpha \bigl(x_{w}^{2}+x_{z}^{2} \bigr)+2(1-\alpha )x_{w}x_{z} \bigr). \end{aligned}$$

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

$$\begin{aligned} \mu _{\alpha } \bigl(G'\bigr)-\mu _{\alpha } (G) \ge & x^{\top }\bigl(D_{\alpha }\bigl(G'\bigr)- D_{\alpha }(G)\bigr)x \\ \ge &\sum_{w\in V(G_{2})}\sum_{z\in V(G_{3})\setminus \{u,v\} } \bigl(\alpha \bigl(x_{w}^{2}+x_{z}^{2} \bigr)+2(1-\alpha )x_{w}x_{z} \bigr) \\ &{}-\sum_{w\in V(G_{1})}\sum_{z\in V(G_{3})\setminus \{u,v\} } \bigl( \alpha \bigl(x_{w}^{2}+x_{z}^{2} \bigr)+2(1-\alpha )x_{w}x_{z} \bigr) \\ =&\varGamma . \end{aligned}$$

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

$$\begin{aligned} 0 =& \mu _{\alpha }\bigl(G'\bigr)x_{v}-\mu _{\alpha }(G)x_{v} \\ =&\sum_{w\in V(G_{3})\setminus \{u,v\}} \bigl(d_{G'}(v,w)-d_{G}(v,w) \bigr) \bigl(\alpha x_{v}+(1-\alpha )x_{w}\bigr) \\ =&\sum_{w\in V(G_{3})\setminus \{u,v\}} \bigl(\alpha x_{v}+(1- \alpha )x_{w}\bigr) \\ >& 0, \end{aligned}$$

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

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

$$\begin{aligned} \mu _{\alpha }\bigl(G''\bigr)-\mu _{\alpha }(G) \ge & x^{\top }\bigl(D_{\alpha } \bigl(G''\bigr)-D_{ \alpha }(G)\bigr)x \\ \ge &\sum_{w\in V(G_{1})}\sum_{z\in V(G_{3})\setminus \{u,v\} } \bigl(\alpha \bigl(x_{w}^{2}+x_{z}^{2} \bigr)+2(1-\alpha )x_{w}x_{z} \bigr) \\ &{}-\sum_{w\in V(G_{2})}\sum_{z\in V(G_{3})\setminus \{u,v\} } \bigl( \alpha \bigl(x_{w}^{2}+x_{z}^{2} \bigr)+2(1-\alpha )x_{w}x_{z} \bigr) \\ =& -\varGamma \\ >&0, \end{aligned}$$

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

LetHbe a connected graph of order at least 3 with\(uv\in E(H)\). Suppose that\(\eta (u)=v\)for some automorphismηofG. 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

$$\begin{aligned} \mu _{\alpha } (G) \le &\mu _{\alpha } \biggl(H_{u,v} \biggl( \frac{p+q+3}{2},\frac{p+q-3}{2} \biggr) \biggr) \\ < &\mu _{\alpha } \biggl(H_{u,v} \biggl(\frac{p+q+1}{2}, \frac{p+q-1}{2} \biggr) \biggr) \\ < & \mu _{\alpha } \biggl(H_{u,v} \biggl(\frac{p+q+3}{2}, \frac{p+q-3}{2} \biggr) \biggr), \end{aligned}$$

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

$$\begin{aligned} \mu _{\alpha } (G) \le & \mu _{\alpha } \biggl(H_{u,v} \biggl( \frac{p+q}{2}+1,\frac{p+q}{2}-1 \biggr) \biggr) \\ < &\mu _{\alpha } \biggl(H_{u,v} \biggl(\frac{p+q}{2}, \frac{p+q}{2} \biggr) \biggr) \\ < & \mu _{\alpha } \biggl(H_{u,v} \biggl(\frac{p+q}{2}-1, \frac{p+q}{2}+1 \biggr) \biggr), \end{aligned}$$

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

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

LetGbe a tree of ordern. 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])

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

$$ \sigma (G)\ge n^{2}-n-4>\sigma \bigl(S_{n}^{+} \bigr)=n^{2}-2n. $$

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

LetGbe 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

$$ \mu _{\alpha }(G)\ge \frac{2\sigma (G)}{n}\ge \frac{2(n^{2}-n-4)}{n}. $$

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

$$ \mu _{\alpha }\bigl(S_{n}^{+}\bigr)< T_{\max }\bigl(S_{n}^{+}\bigr)=2n-3. $$

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

$$ \mu _{\alpha }(G)\ge \frac{2(n^{2}-n-4)}{n}\ge 2n-3>\mu _{\alpha } \bigl(S_{n}^{+}\bigr), $$

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

LetGbe a connected graph of ordernwith 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

LetGbe 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

$$\begin{aligned}& \mu _{\alpha }\bigl(T'\bigr)-\mu _{\alpha }(C_{n}) \\& \quad = x^{\top }\bigl(D_{\alpha }\bigl(T' \bigr)-D_{\alpha }(G)\bigr)x \\& \quad \ge \alpha \bigl(x_{u_{2}}^{2}+x_{u_{1}}^{2} \bigr)+2(1-\alpha )x_{u_{2}}x_{u_{1}} \\& \qquad {} -\sum_{i= \lfloor \frac{n+1}{2} \rfloor +2}^{n} \bigl( \alpha \bigl(x_{u_{2}}^{2}+x_{u_{i}}^{2} \bigr)+2(1-\alpha )x_{u_{2}}x_{u_{i}} \bigr) \\& \qquad {} +\sum_{i=3}^{ \lceil \frac{n+1}{2} \rceil }(n-2i+3) \bigl( \alpha \bigl(x_{u_{2}}^{2}+x_{u_{i}}^{2} \bigr)+2(1-\alpha )x_{u_{2}}x_{u_{i}} \bigr) \\& \quad = 2x_{u_{1}}^{2} \Biggl(1- \biggl(n- \biggl\lfloor \frac{n+1}{2} \biggr\rfloor -1 \biggr)+\sum_{i=3}^{ \lceil \frac{n+1}{2} \rceil }(n-2i+3) \Biggr) \\& \quad = 2x_{u_{1}}^{2} \biggl(1+ \biggl(n-1- \biggl\lceil \frac{n+1}{2} \biggr\rceil \biggr) \biggl( \biggl\lceil \frac{n+1}{2} \biggr\rceil -2 \biggr) \biggr) \\& \quad \ge 2x_{u_{1}}^{2} \\& \quad > 0, \end{aligned}$$

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

LetGbe 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

LetGbe 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

$$\begin{aligned} \varGamma =&\sum_{u\in U_{1} }\sum _{v\in U_{3}} \bigl(\alpha \bigl(x_{u}^{2}+x_{v}^{2} \bigr)+2(1-\alpha )x_{u} x_{v} \bigr) \\ &{}-\sum_{u\in U_{1} }\sum_{v\in U_{2}} \bigl(\alpha \bigl(x_{u}^{2}+x_{v}^{2} \bigr)+2(1-\alpha )x_{u} x_{v} \bigr). \end{aligned}$$

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

$$\begin{aligned} \mu _{\alpha }\bigl(G'\bigr)-\mu _{\alpha }(G) \ge & x^{\top }\bigl(D_{\alpha }\bigl(G' \bigr)-D_{ \alpha }(G)\bigr)x \\ \ge &\sum_{u\in U_{1}}\sum _{v\in U_{3}} \bigl(\alpha \bigl(x_{u}^{2}+x_{v}^{2} \bigr)+2(1-\alpha )x_{u} x_{v} \bigr) \\ &{}-\sum_{u\in V(T_{2k+1})}\sum_{v\in U_{2}} \bigl(\alpha \bigl(x_{u}^{2}+x_{v}^{2} \bigr)+2(1-\alpha )x_{u} x_{v} \bigr) \\ >&\sum_{u\in U_{1}}\sum_{v\in U_{3}} \bigl(\alpha \bigl(x_{u}^{2}+x_{v}^{2} \bigr)+2(1-\alpha )x_{u} x_{v} \bigr) \\ &{}-\sum_{u\in U_{1}}\sum_{v\in U_{2}} \bigl(\alpha \bigl(x_{u}^{2}+x_{v}^{2} \bigr)+2(1-\alpha )x_{u} x_{v} \bigr). \end{aligned}$$

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

$$\begin{aligned} \mu _{\alpha }\bigl(G''\bigr)-\mu _{\alpha }(G) \ge & x^{\top }\bigl(D_{\alpha } \bigl(G''\bigr)-D_{ \alpha }(G)\bigr)x \\ \ge &\sum_{u\in U_{1}}\sum _{v\in U_{2}} \bigl(\alpha \bigl(x_{u}^{2}+x_{v}^{2} \bigr)+2(1-\alpha )x_{u} x_{v} \bigr) \\ &{}-\sum_{u\in V(T_{2k})}\sum_{v\in U_{3}} \bigl(\alpha \bigl(x_{u}^{2}+x_{v}^{2} \bigr)+2(1-\alpha )x_{u} x_{v} \bigr) \\ >&\sum_{u\in U_{1}}\sum_{v\in U_{2}} \bigl(\alpha \bigl(x_{u}^{2}+x_{v}^{2} \bigr)+2(1-\alpha )x_{u} x_{v} \bigr) \\ &{}-\sum_{u\in U_{1}}\sum_{v\in U_{3}} \bigl(\alpha \bigl(x_{u}^{2}+x_{v}^{2} \bigr)+2(1-\alpha )x_{u} x_{v} \bigr) \\ >& 0. \end{aligned}$$

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 \(\alpha =\frac{1}{2}\).

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

$$ \lambda _{j}(A+B)\le \lambda _{i}(A)+\lambda _{j-i+1}(B)\quad \mbox{for }1\le i\le j\le n, $$

and

$$ \lambda _{j}(A+B)\ge \lambda _{i}(A)+\lambda _{j-i+n}(B) \quad \mbox{for }1\le j\le i\le n. $$

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

Theorem 6.1

LetGbe a connected graph andλbe any eigenvalue of\(D_{\alpha }(G)\)other than the distanceα-spectral radius. Then

$$ 2\alpha T_{\min }(G)-T_{\max }(G)+(1-\alpha ) (n-2)\le \lambda \le T_{ \max }(G)-(1-\alpha )n. $$

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

$$\begin{aligned} \lambda _{j}\bigl(D_{\alpha }(G)\bigr) \le & \lambda _{1}(B)+\lambda _{j}(A) \\ \le & T_{\max }(G)-\alpha T_{\min }(G)-(1-\alpha ) (n-1)+ \alpha T_{ \min }(G)-1+\alpha \\ =&T_{\max }(G)-(1-\alpha )n \end{aligned}$$

and

$$\begin{aligned} \lambda _{j}\bigl(D_{\alpha }(G)\bigr) \ge & \lambda _{n}(B)+\lambda _{j}(A) \\ \ge & -T_{\max }(G)+\alpha T_{\min }(G)+(1-\alpha ) (n-1)+ \alpha T_{ \min }(G)-1+\alpha \\ =& 2\alpha T_{\min }(G)-T_{\max }(G)+(1-\alpha ) (n-2). \end{aligned}$$

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

$$ \vert \lambda \vert \le T_{\max }(G)-(1-\alpha ) (n-2). $$

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

$$ \mathcal{E}_{\alpha }(G)=\sum_{i=1}^{n} \biggl\vert \mu _{\alpha }^{(i)}(G)- \frac{2\alpha \sigma (G)}{n} \biggr\vert . $$

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

$$ \mathcal{E}_{1/2}(G)=\frac{1}{2}\sum _{i=1}^{n} \biggl\vert 2\mu _{1/2}^{(i)}(G)- \frac{2\sigma (G)}{n} \biggr\vert $$

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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Acknowledgements

We would like to thank the referees and the editor for their suggestions and comments.

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This work was supported by the National Natural Science Foundation of China (No. 11671156).

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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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MSC

  • 05C50
  • 05C12

Keywords

  • Distance spectral radius
  • Distance signless Laplacian spectral radius
  • Local graft transformation
  • Extremal graph