On the sum of powers of the $A_{\alpha}$-eigenvalues of graphs

1.   Introduction

Let $G$ be a simple finite undirected connected graph with vertex set $V(G)$ and edge set $E(G)$, where $|V(G)|$ is the order and $|E(G)|$ is the size of $G$. Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree diagonal matrix of a graph $G$, respectively. Then $L(G) = D(G)-A(G)$, $Q(G) = D(G)+A(G)$ and $\mathcal{L}(G) = D^{-\frac{1}{2}}(G)L(G)D^{-\frac{1}{2}}(G)$ are called the Laplacian matrix, the signless Laplacian matrix and the normalized Laplacian matrix of the graph $G$, respectively.

The investigation on the sum of the $p$-th power of the eigenvalues of graphs is a topic of interest in Mathematical Chemistry. Based on the mathematical methods, scholars get many bounds for the sum of the $p$-th power of the eigenvalues of graphs. For a non-zero real number $p$, Zhou ^[1] introduced the sum of the $p$-th power of the non-zero Laplacian eigenvalues of $G$, denoted by $S_{L}^p(G)$. Since $S_{L}^p(G)$ has close relation with the Laplacian-energy-like invariant ^[2], the Laplacian Estrada index ^[3] and the Kirhhoff index ^[4], there are considerable results regarding $S_{L}^p(G)$ in the literature. For related results, one may refer to ^{[1,5,6,7,8,9]} and references therein. For a non-zero real number $p$, M. Liu and B. Liu ^[10] defined $S_{Q}^p(G)$ as the sum of the $p$-th power of the non-zero signless Laplacian eigenvalues of $G$, which has close relation with the incidence energy ^[11] and the signless Laplacian Estrada index ^[12]. For details on $S_{Q}^p(G)$, see the papers ^[13,14] and the references cited therein. Moreover, Akbari et al. ^[15,16] compared between $S_{L}^p(G)$ and $S_{Q}^p(G)$ when the parameter $p$ takes different values. For a non-zero real number $p$, Ş.B. Bozkurt and D. Bozkurt ^[17] defined $S_{\mathcal{L}}^p(G)$ as the sum of the $p$-th power of the normalized Laplacian eigenvalues of $G$, which has close relation with the degree-Kirchhoff index ^[18] and the general Randić index ^[19]. For related results, one may refer to ^[20,21].

For any real number $\alpha \in[0, 1]$, Nikiforov ^[22] defined the $A_{\alpha}$-matrix of $G$ as

It is easy to see that $A_{\alpha}(G)$ is the adjacency matrix $A(G)$ if $\alpha = 0$, and $A_{\alpha}(G)$ is essentially equivalent to signless Laplacian matrix $Q(G)$ if $\alpha = \frac{1}{2}$. The new matrix $A_{\alpha}(G)$ not only can underpin a unified theory of $A(G)$ and $Q(G)$, but it also brings many new interesting problems, see for example ^{[22,23,24,25]}. In particular, $A_{\alpha}(G)$ is a positive definite matrix for $\frac{1}{2} < \alpha < 1$, which is a hitherto uncharted territory of worth our investigation and exploration, see ^[22]. Moreover, X. Liu and S. Liu ^[26] found that $A_{\alpha}$-eigenvalues (especially, $\frac{1}{2} < \alpha < 1$) are much more efficient than $A$-eigenvalues and $Q$-eigenvalues when we use them to distinguish graphs, by enumerating the $A_{\alpha}$-characteristic polynomials for all graphs on at most ten vertices. The $A_{\alpha}$-matrix has been an interesting topic in mathematical literature and has been studied extensively, see for example ^{[22,23,27,28,29,30]} and references therein.

Let $\lambda_1(A_{\alpha}(G))\geq \lambda_2(A_{\alpha}(G))\geq \cdots \geq \lambda_n(A_{\alpha}(G))$ be the $A_{\alpha}$-eigenvalues of a graph $G$ of order $n$. Motivated by the above work, we define $S_{\alpha}^p(G)$ as the sum of the $p$-th power of the $A_{\alpha}$-eigenvalues of $G$, that is,

where $\frac{1}{2} < \alpha < 1$ and $p$ is a real number. $S_{\alpha}^p(G)$ can be regard as a generalization of $S_{Q}^p(G)$ due to the fact that our results are correct for the sum of the $p$-th power of the non-zero $A_{\frac{1}{2}}$-eigenvalues of $G$. By using the Maclaurin development, we have

where $p$ is an integer and $E_{\alpha}(G)$ is called the $\alpha$-Estrada index defined by Cardoso et al. ^[31]. Thus the bound for $S_{\alpha}^p(G)$ can be naturally converted to the bound of the $\alpha$-Estrada index. In addition, we find that $S_{\alpha}^p(G)$ is connected with the first general Zagreb index, which is a useful topological index and has important applications in chemistry.

The primary purpose of this paper is to establish the bounds of $S_{\alpha}^p(G)$. The cases $p = 0$ and $p = 1$ are trivial as $S_{\alpha}^{0}(G) = n$ and $S_{\alpha}^{1}(G) = 2\alpha m$, where $m$ is the size of $G$. We will not consider both cases in the following results. The rest of the paper is organized as follows. In Section 2, we recall some useful notions and lemmas used further. In Section 3, some bounds on the $S_{\alpha}^p(G)$ are presented. In Sections 4 and 5, several bounds for $S_{\alpha}^{p}(G)$ related to degree sequences, order and size are given through majorization techniques. In Section 6, lower and upper bounds for $S_{\alpha}^{p}(G)$ of a bipartite graph $G$ are obtained, and the extremal graphs characterized.

2.   Preliminaries

Let $G-e$ denote the graph that arises from $G$ by deleting the edge $e\in E(G)$. A connected graph is called a $c$-cyclic graph if it contains $n$ vertices and $n+c-1$ edges. For $v_i\in V(G)$, $d_G(v_i) = d_i(G)$ denotes the degree of vertex $v_i$ in $G$. The minimum and the maximum degree of $G$ are denoted by $\delta = \delta(G)$ and $\Delta = \Delta(G)$, respectively. A pendant vertex is a vertex of degree one and a quasi-pendant vertex is a vertex adjacent to a pendant vertex. Li and Zheng ^[32] defined the first general Zagreb index of $G$ as $Z_p = Z_p(G) = \sum_{v\in V(G)}d^p(v)$, where $p$ is an arbitrary real number except 0 and 1. A subset $I$ of $V(G)$ is called an independent set of a graph $G$ if no two vertices in $I$ are adjacent in $G$. Given a graph $G$, the independence number $\theta(G)$ of $G$ is the numbers of vertices of the largest independent set. Denote by $K_n$, $K_{a, \, b}$ and $\overline{G}$ the complete graph, the complete bipartite graph and the complement of a graph $G$, respectively. The join $G_1\vee G_2$ of two vertex-disjoint graphs $G_1$ and $G_2$ is the graph formed from the union of $G_1$ and $G_2$ by joining each vertex of $G_1$ to each vertex of $G_2$.

Lemma 2.1. (^[33]) Let $G$ be a graph with $n$ vertices. If $e\in E(G)$ and $\frac{1}{2}\leq \alpha \leq 1$, then$\lambda_i(A_{\alpha}(G))\geq \lambda_i(A_{\alpha}(G-e))$for $1\leq i \leq n$.

Lemma 2.2. (^[34,35]) Let $G$ be a graph of order $n$ and size $m$. Then

with equality if and only if $G$ has the property $d_2 = d_3 = \cdots = d_{n-1} = \frac{\Delta+\delta}{2}$, which includes also the regular graphs.

Lemma 2.3. (^[36]) Let $s_1, s_2, \ldots, s_n$ be the singular values of a matrix $M = (m_{ij})\in M_n$. Then

Lemma 2.4. (^[37]) For $c$-cyclic graphs with $n$ vertices, the minimal degree sequences with respect to the majorization order are given by $(2, 2, \ldots, 2, 1, 1)$, in case $c = 0$ and $n > 2$, $(2, 2, \ldots, 2)$, in case $c = 1$ and $n > 2$, $(\underbrace{3, 3, \ldots, 3}_{2c-2}, 2, 2, \ldots, 2)$, in case $2\leq c \leq 6$ and $n > 2c-2$.

Lemma 2.5. (^[38]) Let $a(G)$, $b(G)$ and $m_G(\alpha)$ be the number of pendant vertices, quasi-pendant vertices of $G$ and the multiplicity of $\alpha$ as an eigenvalue of $A_{\alpha}(G)$, respectively. Then$m_G(\alpha)\geq a(G)-b(G)$with equality if each internal vertex is a quasi-pendant vertex.

Lemma 2.6. (^[39]) Let $G$ be a graph of order $n$ and size $m$. If $\alpha\in (\frac{1}{2}, 1)$, then

the equality holds if and only if $G\cong K_n$.

3.   Some bounds on $S_{\alpha}^{p}(G)$

Theorem 3.1. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph with $n$ vertices and $e\in E(G)$.

(i) If $p > 0$ and $p\neq 1$, then$S_{\alpha}^{p}(G-e) < S_{\alpha}^{p}(G)$.

(ii) If $p < 0$, then$S_{\alpha}^{p}(G-e) > S_{\alpha}^{p}(G)$.

Proof. By Perron-Frobenius Theorem, we have $\lambda_1(A_{\alpha}(G)) > \lambda_1(A_{\alpha}(G-e))$. By Lemma 2.1, the result follows.

Corollary 3.1. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$.

(i) If $p > 0$ and $p\neq 1$, then

with equality if and only if $G\cong K_n$.

(ii) If $p < 0$, then

with equality if and only if $G\cong K_n$.

Proof. From Proposition 36 in ^[22], it follows that $\lambda_1(A_{\alpha}(K_n)) = n-1$ and $\lambda_i(A_{\alpha}(K_n)) = \alpha n-1$ for $2\leq i\leq n$. By Theorem 3.1, we have the proof.

Corollary 3.2. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$ with independence number $\theta$.

(i) If $p > 0$ and $p\neq 1$, then

where $x_1$ and $x_2$ are the roots of the equation

and equality holds if and only if $G\cong \overline{K_{\theta}}\vee K_{n-\theta}$.

(ii) If $p < 0$, then

where $x_1$ and $x_2$ are the roots of the equation

and equality holds if and only if $G\cong \overline{K_{\theta}}\vee K_{n-\theta}$.

Proof. Let $\phi_{\alpha}(G, x)$ be the characteristic polynomial of $A_{\alpha}(G)$. By direct computation, we have

Thus

where $x_1$ and $x_2$ are the roots of the equation $x^2-(\alpha n+n-\theta-1)x+\alpha\theta+\alpha n^2-\alpha n-\alpha \theta^2-\theta n+\theta^2 = 0$. By Theorem 3.1, we have the proof.

Theorem 3.2. Let $G$ be a connected graph of order $n$ and size $m$. If $\frac{1}{2} < \alpha < 1$ and $p\neq0$ and $p\neq1$, then

with equality if and only if $G\cong K_n$.

Proof. By the arithmetic-geometric mean inequality, we have

Let $h(x) = x^p+(n-1)\left(\frac{\det(A_{\alpha}(G))}{x}\right)^{\frac{p}{n-1}}$. Then $h'(x) = p(x^{p-1}-\det(A_{\alpha}(G))^{\frac{p}{n-1}}x^{-\frac{p}{n-1}-1})$. It is easy to see that $h(x)$ is increasing on $[\det(A_{\alpha}(G))^{\frac{1}{n}}, +\infty)$ whether $p > 0$ or $p < 0$. From Corollary 19 in ^[22], it follows that

Thus

with equality if and only if $\lambda_1(A_{\alpha}(G)) = \frac{2m}{n}$ and $\lambda_2(A_{\alpha}(G)) = \cdots = \lambda_n(A_{\alpha}(G))$. From Corollary 33 in ^[22], the diameter of $G$ is $1$. Thus, $G\cong K_n$. Conversely, if $G\cong K_n$, then $\lambda_1(A_{\alpha}(G)) = n-1$, and $\lambda_i(A_{\alpha}(G)) = \alpha n-1$ for $2\leq i\leq n$. It is easy to check that equality holds in (3.1). This completes the proof.

Theorem 3.3. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$ and size $m$.

(i) If $p < 0$ or $p > 1$, then

with equality if and only if $G\cong K_n$.

(ii) If $0 < p < 1$, then

with equality if and only if $G\cong K_n$.

Proof. Since $p < 0$ or $p > 1$, we know that $f(x) = x^p$ is a strictly convex function. By Jensen's inequality, we have

that is,

Thus

Let $g(x) = x^p+\frac{1}{(n-1)^{p-1}}(2\alpha m-x)^p$. Then $g'(x) = p\left(x^{p-1}-\frac{(2\alpha m-x)^{p-1}}{(n-1)^{p-1}}\right)\geq 0$ for $x\geq \frac{2\alpha m}{n}$. Hence $g(x)$ is increasing on $[\frac{2\alpha m}{n}, +\infty)$. From Lemma 2.2 and Corollary 19 in ^[22], it follows that $\lambda_1(A_{\alpha}(G)) \geq \sqrt{\frac{Z_2}{n}}\geq \frac{2\alpha m}{n}$. Thus

with equality if and only if $\lambda_1(A_{\alpha}(G)) = \sqrt{\frac{Z_2}{n}}$ and $\lambda_2(A_{\alpha}(G)) = \cdots = \lambda_n(A_{\alpha}(G))$. From Corollary 33 in ^[22], the diameter of $G$ is $1$. Thus, $G\cong K_n$. Conversely, if $G\cong K_n$, then $\lambda_1(A_{\alpha}(G)) = n-1$, and $\lambda_i(A_{\alpha}(G)) = \alpha n-1$ for $2\leq i\leq n$. It is easy to check that equality holds in (3.2).

Now suppose that $0 < p < 1$. Then

with equality if and only if $\lambda_2(A_{\alpha}(G)) = \cdots = \lambda_n(A_{\alpha}(G))$, and $g(x)$ is decreasing on $[\frac{2\alpha m}{n}, +\infty)$. By similar arguments as above, the second part of the theorem follows.

Combining the above arguments, we have the proof.

By Lemma 2.2 and Theorem 3.3, we have

Corollary 3.3. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$ and size $m$.

(i) If $p < 0$ or $p > 1$, then

with equality if and only if $G\cong K_n$.

(ii) If $0 < p < 1$, then

with equality if and only if $G\cong K_n$.

Theorem 3.4. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$ and size $m$.

(i) If $0 < p\leq 2$, then$S_{\alpha}^{p}(G)\leq \alpha^pZ_p+2m(1-\alpha)^p$.

(ii) If $p > 2$, then$S_{\alpha}^{p}(G)\geq \alpha^pZ_p+2m(1-\alpha)^p$.

Proof. Since $A_{\alpha}(G)$ is a real symmetric and positive definite matrix for $\frac{1}{2} < \alpha < 1$, the singular values of $A_{\alpha}(G)$ are equal to the eigenvalues of $A_{\alpha}(G)$. By Lemma 2.3, we have the proof.

4.   Bounds for $S_{\alpha}^{p}(G)$ related to degree sequences

Suppose $x = (x_1, x_2, \ldots, x_n)$ and $y = (y_1, y_2, \ldots, y_n)$ are two non-increasing sequences of real numbers, we say $x$ is majorized by $y$, denoted by $x\preceq y$, if $\sum\limits_{i = 1}^{n}x_i = \sum\limits_{i = 1}^{n}y_i$ and $\sum\limits_{i = 1}^{j}x_i\leq \sum\limits_{i = 1}^{j}y_i$ for $j = 1, 2, \ldots, n-1$. For a real-valued function $f$ defined on a set in $\mathbb{R}^n$, if $f(x)\leq f(y)$ whenever $x\preceq y$ but $x\neq y$, then $f$ is said to be Schur-convex.

Theorem 4.1. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$ and size $m$ with the degree sequence $d_1\geq d_2\geq \cdots\geq d_n$.

(i) If $p < 0$ or $p > 1$, then

(ii) If $0 < p < 1$, then

Proof. Let $x = \frac{2\alpha m}{n}(1, \ldots, 1)$, $y = (\alpha d_1, \ldots, \alpha d_n)$ and $z = (\lambda_1(A_{\alpha}(G)), \ldots, \lambda_n(A_{\alpha}(G)))$. It is well known that the spectrum of any symmetric, positive semi-definite matrix majorizes its main diagonal ^[40], hence $x\preceq y\preceq z$. Since $p < 0$ or $p > 1$, $f(x) = x^p$ is a convex function. From ^[41], we know that if the real-valued function $f$ defined on an interval in $\mathbb{R}$ is a convex then $\sum\limits_{i = 1}^{n}f(x_i)$ is Schur-convex. Thus $\frac{(2\alpha m)^p}{n^{p-1}}\leq \sum\limits_{i = 1}^{n}\alpha^p d_i^p\leq S_{\alpha}^{p}(G)$.

If $0 < p < 1$, then $g(x) = -x^p$ is a convex function. By similar arguments as above, the second part of the theorem follows.

By Lemma 2.4 and Theorem 4.1, we have

Corollary 4.1. Let $\frac{1}{2} < \alpha < 1$, $0\leq c \leq 6$ and $G$ be a $c$-cyclic graph with $n$ vertices.

(i) If $p < 0$ or $p > 1$, $c = 0$ and $n > 2$, then

If $p < 0$ or $p > 1$, $c = 1$ and $n > 2$, then

If $p < 0$ or $p > 1$, $2\leq c \leq 6$ and $n > 2c-2$, then

(ii) If $0 < p < 1$, $c = 0$ and $n > 2$, then

If $0 < p < 1$, $c = 1$ and $n > 2$, then$S_{\alpha}^{p}(G)\leq n(2\alpha)^p$.

If $0 < p < 1$, $2\leq c \leq 6$ and $n > 2c-2$, then

Theorem 4.2. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$ and size $m$ with the degree sequence $d_1\geq d_2\geq \cdots\geq d_n = \delta$.

(i) If $p < 0$ or $p > 1$, then

(ii) If $0 < p < 1$, then

Proof. Let $x = (\lambda_1(A_{\alpha}(G)), \lambda_2(A_{\alpha}(G)), \ldots, \lambda_n(A_{\alpha}(G)))$ and $y = (\alpha d_1+(1-\alpha)(n-1), \alpha d_2+(1-\alpha)(n-2), \ldots, \alpha d_{n-1}+(1-\alpha), 2\alpha m-\alpha(2m-\delta)-(1-\alpha)\frac{n(n-1)}{2})$. From Theorem 3.1 in ^[28], it follows that $\lambda_i(A_{\alpha}(G))\leq \alpha d_i+(1-\alpha)(n-i)$ for $1\leq i \leq n$. Thus $x\preceq y$. Similar to the method used in Theorem 4.1, we have the proof.

Theorem 4.3. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$ and size $m$ with the degree sequence $\Delta = d_1\geq d_2\geq \cdots\geq d_n$.

(i) If $p < 0$ or $p > 1$ and $d_2\geq d_3\geq \cdots \geq d_k\geq \alpha n-1$, then

where $2\leq k\leq n$.If $p < 0$ or $p > 1$ and $d_2\leq \alpha n-1$, then

where $2\leq k\leq n$.

(ii) If $0 < p < 1$ and $d_2\geq d_3\geq \cdots \geq d_k\geq \alpha n-1$, then

where $2\leq k\leq n$.If $0 < p < 1$ and $d_2\leq \alpha n-1$, then

where $2\leq k\leq n$.

Proof. Let $x = (\lambda_1(A_{\alpha}(G)), \lambda_2(A_{\alpha}(G)), \ldots, \lambda_n(A_{\alpha}(G)))$ and $y = (d_1, \alpha n-1, \ldots, \alpha n-1, 2\alpha m-d_1-(k-1)(\alpha n-1), 0, \ldots, 0)$. From Proposition 10 in ^[22], it follows that $\lambda_1(A_{\alpha}(G))\leq d_1$. By Lemma 2.1, we have $\lambda_{i}(A_{\alpha}(G))\leq \lambda_{i}(A_{\alpha}(K_n)) = \alpha n-1$ for $2\leq i\leq n$. Thus $x\preceq y$. Similar to the method used in Theorem 4.1, we have the proof.

Remark 4.1. It is easy to see that the equality in (4.1) and (4.2) holds if $G\cong K_n$.

Theorem 4.4. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$ and size $m$ with the degree sequence $\Delta = d_1\geq d_2\geq \cdots\geq d_n$.

(i) If $p < 0$ or $p > 1$ and $n-1 > d_1\geq d_2\geq d_3\geq \cdots \geq d_k\geq \alpha (n-2)$, then

where $2\leq k\leq n$.If $p < 0$ or $p > 1$, $d_1 < n-1$ and $d_2\leq \alpha(n-2)$, then

where $2\leq k\leq n$.

(ii) If $0 < p < 1$ and

then

where $2\leq k\leq n$.If $0 < p < 1$, $d_1 < n-1$ and $d_2\leq \alpha(n-2)$, then

where $2\leq k\leq n$.

Proof. Let $x = (\lambda_1(A_{\alpha}(G)), \lambda_2(A_{\alpha}(G)), \ldots, \lambda_n(A_{\alpha}(G)))$ and $y = (d_1, \alpha(n-2), \ldots, \alpha(n-2), 2\alpha m-d_1-\alpha(k-1)(n-2), 0, \ldots, 0)$. From Proposition 10 in ^[22] and Theorem 3.1 in ^[27], it follows that $\lambda_1(A_{\alpha}(G))\leq d_1$ and $\lambda_{i}(A_{\alpha}(G))\leq \alpha(n-2)$ for $2\leq i\leq n$. Thus $x\preceq y$. Similar to the method used in Theorem 4.1, we have the proof.

Theorem 4.5. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$ and size $m$ with the degree sequence $\Delta = d_1\geq d_2\geq \cdots\geq d_n$.

(i) If $p < 0$ or $p > 1$ and $m_G(\lambda_j)$ is the multiplicity of $\lambda_j$ as an eigenvalue of $A_{\alpha}(G)$, then

where $2\leq k\leq j\leq n$.

(ii) If $0 < p < 1$ and $m_G(\lambda_j)$ is the multiplicity of $\lambda_j$ as an eigenvalue of $A_{\alpha}(G)$, then

where $2\leq k\leq j\leq n$.

Proof. Let $x = (\lambda_1(A_{\alpha}(G)), \ldots, \lambda_n(A_{\alpha}(G)))$ and $y = (d_1, d_2, \ldots, d_k, \lambda_j, \ldots, \lambda_j, 2\alpha m-d_1-\sum\limits_{i = 2}^{k}d_k-m_G(\lambda_j)\lambda_j, 0, \ldots, 0)$. From Proposition 10 in ^[22], it follows that $\lambda_i(A_{\alpha}(G))\leq d_i$ for $1\leq i \leq n$. Then $x\preceq y$. Similar to the method used in Theorem 4.1, we have the proof.

By Lemma 2.5 and Theorem 4.5, we have

Corollary 4.2. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$ and size $m$ with the degree sequence $\Delta = d_1\geq d_2\geq \cdots\geq d_n$, and let $a$ and $b$ be the number of pendant vertices and quasi-pendant vertices of $G$, respectively.

(i) If $p < 0$ or $p > 1$ and $a-b\geq 1$, then

(ii) If $0 < p < 1$ and $a-b\geq 1$, then

5.   Bounds for $S_{\alpha}^{p}(G)$ related to order and size

Theorem 5.1. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$ and size $m$.

(i) If $p < 0$ or $p > 1$ and there is $c$ such that $\lambda_1(A_{\alpha}(G))\geq c > 0$, then

(ii) If $0 < p < 1$ and there is $c$ such that $\lambda_1(A_{\alpha}(G))\geq c > 0$, then

Proof. Let $x = (c, \frac{2\alpha m-c}{n-1}, \ldots, \frac{2\alpha m-c}{n-1})$ and

Since $\lambda_1(A_{\alpha}(G))\geq c > 0$, we have $x\preceq y$. Similar to the method used in Theorem 4.1, we have the proof.

Corollary 5.1. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$ and size $m$.

(i) If $p < 0$ or $p > 1$, then

(ii) If $0 < p < 1$, then

Proof. From Corollary 13 in ^[22], it follows that

By Theorem 5.1, we have the proof.

Corollary 5.2. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$ and size $m$.

(i) If $p < 0$ or $p > 1$, then

(ii) If $0 < p < 1$, then

Proof. From Corollary 19 in ^[22], it follows that

By Theorem 5.1, we have the proof.

Remark 5.1. It is easy to see that the equality in Corollary 5.2 holds if $G\cong K_n$.

Corollary 5.3. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$ and size $m$ with chromatic number $\chi$.

(i) If $p < 0$ or $p > 1$, then

(ii) If $0 < p < 1$, then

The equality holds in (5.3) and (5.4) if and only if $G\cong K_n$.

Proof. It is well known that $\lambda_1(A(G))\geq \chi-1$ with equality if and only if $G$ is a complete graph or an odd cycle, see ^[42]. From Proposition 18 in ^[22], it follows that

with equality if and only if $G$ is a complete graph or an odd cycle. By Theorem 5.1, we have the proof.

Theorem 5.2. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected graph of order $n$ and size $m$.

(i) If $p < 0$ or $p > 1$, then

(ii) If $0 < p < 1$, then

The equality holds in (5.5) and (5.6) if and only if $G\cong K_n$.

Proof. Let $x = (\lambda_1(A_{\alpha}(G)), \ldots, \lambda_n(A_{\alpha}(G)))$ and $y = (\frac{2m}{n-1}(1-\alpha)+\alpha n-1, \alpha n-1, \ldots, \alpha n-1, 2\alpha m-\frac{2m}{n-1}(1-\alpha)-(n-1)(\alpha n-1))$. By Lemmas 2.1 and 2.6, we have

and

for $2\leq i\leq n$. Thus $x\preceq y$. Similar to the method used in Theorem 4.1, we have the proof.

6.   $S_{\alpha}^{p}(G)$ of bipartite graphs

Theorem 6.1. Let $\frac{1}{2} < \alpha < 1$, $G$ be a connected bipartite graph with $n$ vertices.

(i) If $p > 1$, then $S_{\alpha}^{p}(G)\leq S_{\alpha}^{p}(K_{1, \, n-1})$ with equality if and only if $G\cong K_{1, \, n-1}$.

(ii) If $0 < p < 1$, then $S_{\alpha}^{p}(G)\leq S_{\alpha}^{p}(K_{\lceil n/2\rceil, \, \lfloor n/2\rfloor})$ with equality if and only if $G\cong K_{\lceil n/2\rceil, \, \lfloor n/2\rfloor}$.

(iii) If $p < 0$, then $S_{\alpha}^{p}(G)\geq S_{\alpha}^{p}(K_{\lceil n/2\rceil, \, \lfloor n/2\rfloor})$ with equality if and only if $G\cong K_{\lceil n/2\rceil, \, \lfloor n/2\rfloor}$.

Proof. Let $G = (X, Y)$ be a connected bipartite graph on $n$ vertices and suppose that $|X| = a$, $|Y| = b$ and $a\geq b\geq 1$, where $a+b = n$. From Proposition 38 in ^[22], it follows that

Let

Then

If $p > 1$, then $f(x)$ is decreasing for $1\leq x \leq \frac{n}{2}$ and increasing for $\frac{n}{2}\leq x \leq n-1$. Hence $f(n/2)\leq f(x)\leq f(1)$. By Theorem 3.1, we have $S_{\alpha}^{p}(G) < S_{\alpha}^{p}(K_{a, \, b})\leq S_{\alpha}^{p}(K_{1, \, n-1})$ for $p > 0$, $p\neq 1$ and $G\neq K_{a, \, b}$.

If $0 < p < 1$, then $f(x)$ is increasing for $1\leq x \leq \frac{n}{2}$ and decreasing for $\frac{n}{2}\leq x \leq n-1$. Hence $f(1)\leq f(x)\leq f(n/2)$. By Theorem 3.1, we have $S_{\alpha}^{p}(G) < S_{\alpha}^{p}(K_{a, \, b})\leq S_{\alpha}^{p}(K_{\lceil n/2\rceil, \, \lfloor n/2\rfloor})$ for $p > 0$, $p\neq 1$ and $G\neq K_{a, \, b}$.

If $p < 0$, then $f(x)$ is decreasing for $1\leq x \leq \frac{n}{2}$ and increasing for $\frac{n}{2}\leq x \leq n-1$. Hence $f(n/2)\leq f(x)\leq f(1)$. By Theorem 3.1, we have $S_{\alpha}^{p}(G) > S_{\alpha}^{p}(K_{a, \, b})\geq S_{\alpha}^{p}(K_{\lceil n/2\rceil, \, \lfloor n/2\rfloor})$ for $p < 0$ and $G\neq K_{a, \, b}$.

Combining the above arguments, we have the proof.

Acknowledgment

The author is grateful to the anonymous referee for careful reading and valuable comments which result in an improvement of the original manuscript. This work was supported by the National Natural Science Foundation of China (No. 12071411).

Conflict of interest

The authors declared that they have no conflicts of interest to this work.