ABSTRACT

Let G be a graph with order n and size m. For each real number $\alpha \in [0,1]$, the $A_{\alpha }$-matrix of a graph G is defined as $A_{\alpha }\left(G\right):=\alpha D\left(G\right)+(1-\alpha )A\left(G\right)$, where $D\left(G\right)=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(d_1,\dots ,d_n)$ is the degree matrix of G, and $A\left(G\right)$ is its adjacency matrix. The functions $s\left(G\right)=\sum _{1\le i\le n}\left|d_i-\frac{2m}{n}\right|\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{v}\mathrm{a}\mathrm{r}\left(G\right)=\frac{1}{n}\sum _{1\le i\le n}{\left(d_i-\frac{2m}{n}\right)}^2$have been already used elsewhere as measures of graph irregularity. In this paper, we find several bounds for the spectral radius $\lambda \left(G\right)$ of the matrix $A_{\alpha }\left(G\right)$. Especially, when G is non-empty we prove for $\alpha \in [0,1)$ that $\begin{array}{rl}& \frac{s^2\left(G\right)}{2n^2\sqrt{n(2\alpha -{\alpha }^2){\mathrm{\Delta }}^2+2(1-\alpha {)}^{2}m}}\\ & \le \lambda \left(G\right)-\frac{2m}{n}\le 1+\sqrt{(2\alpha -{\alpha }^2)n\mathrm{\Omega }(G{)}^2+(1-\alpha {)}^{2}s\left(G\right)},\end{array}$where $\mathrm{\Omega }\left(G\right)=\mathrm{\Delta }\left(G\right)-\lceil 2m/n\rceil +1$ and $\mathrm{\Delta }\left(G\right)$ is the maximum vertex degree of G. These inequalities extend to the $A_{\alpha }$-matrix a Nikiforov's result involving the spectral radius of the adjacency matrix. In addition, for fixed order and size, we determine the graphs minimizing the degree variance $\mathrm{v}\mathrm{a}\mathrm{r}\left(G\right)$ and compute the $A_{\alpha }$-spectrum of some blown-up graphs.

摘要

令G为阶数为n且大小为m的图。对于每个实数$\alpha \in [0,\mathrm{1个}]$， 这${\mathrm{一种}}_{\alpha }$- 图G的矩阵定义为${\mathrm{一种}}_{\alpha }\left(G\right):=\alpha 丁\left(G\right)+(\mathrm{1个}-\alpha )\mathrm{一种}\left(G\right)$， 在哪里$丁\left(G\right)=\mathrm{d}\mathrm{一世}\mathrm{一种}\mathrm{G}(d_{\mathrm{1个}},\dots \dots ,d_n)$是G的度矩阵，并且$\mathrm{一种}\left(G\right)$是它的邻接矩阵。功能$秒\left(G\right)=\sum _{\mathrm{1个}\le \mathrm{一世}\le n}\left|d_{\mathrm{一世}}-\frac{\mathrm{2个}米}{n}\right|\mathrm{一种}\mathrm{n}\mathrm{d}\mathrm{v}\mathrm{一种}\mathrm{r}\left(G\right)=\frac{\mathrm{1个}}{n}\sum _{\mathrm{1个}\le \mathrm{一世}\le n}{\left(d_{\mathrm{一世}}-\frac{\mathrm{2个}米}{n}\right)}^{\mathrm{2个}}$已经在其他地方用作图形不规则性的度量。在本文中，我们找到了光谱半径的几个界限$\lambda \left(G\right)$矩阵的${\mathrm{一种}}_{\alpha }\left(G\right)$. 特别是，当G非空时，我们证明$\alpha \in [0,\mathrm{1个})$那$\begin{array}{rl}& \frac{{秒}^{\mathrm{2个}}\left(G\right)}{\mathrm{2个}{n}^{\mathrm{2个}}\sqrt{n(\mathrm{2个}\alpha -{\alpha }^{\mathrm{2个}}){\mathrm{△}}^{\mathrm{2个}}+\mathrm{2个}(\mathrm{1个}-\alpha {)}^{\mathrm{2个}}米}}\\ & \le \lambda \left(G\right)-\frac{\mathrm{2个}米}{n}\le \mathrm{1个}+\sqrt{(\mathrm{2个}\alpha -{\alpha }^{\mathrm{2个}})n\mathrm{欧姆}(G{)}^{\mathrm{2个}}+(\mathrm{1个}-\alpha {)}^{\mathrm{2个}}秒\left(G\right)},\end{array}$在哪里$\mathrm{欧姆}\left(G\right)=\mathrm{△}\left(G\right)-\lceil \mathrm{2个}米/n\rceil +\mathrm{1个}$和$\mathrm{△}\left(G\right)$是G的最大顶点度。这些不平等延伸到${\mathrm{一种}}_{\alpha }$-matrix 涉及邻接矩阵谱半径的 Nikiforov 结果。此外，对于固定顺序和大小，我们确定最小化度方差的图$\mathrm{v}\mathrm{一种}\mathrm{r}\left(G\right)$并计算${\mathrm{一种}}_{\alpha }$-一些放大图的频谱。