ABSTRACT

Let D be a digraph of order n and let $A\left(D\right)$ be the adjacency matrix of D. Let $Deg\left(D\right)$ be the diagonal matrix of vertex out-degrees of D. For any real $\alpha \in [0,1]$, the generalized adjacency matrix $A_{\alpha }\left(D\right)$ of the digraph D is defined as $A_{\alpha }\left(D\right)=\alpha Deg\left(D\right)+(1-\alpha )A\left(D\right).$ The largest modulus of the eigenvalues of $A_{\alpha }\left(D\right)$ is called the generalized adjacency spectral radius or the $A_{\alpha }$-spectral radius of D. In this paper, we obtain some lower bounds for the spectral radius of $A_{\alpha }\left(D\right)$ in terms of the number of vertices, the number of arcs and the number of closed walks of the digraph D. The extremal graphs attained these lower bounds are determined. We also obtain some bounds for the spectral radius of $A_{\alpha }\left(D\right)$ in terms of the vertex out-degrees, the vertex average 2-out-degrees of the vertices of D and parameter α. We characterize the extremal digraphs attaining these bounds.

摘要

令D为n阶有向字母并令$\mathrm{一个}\left(丁\right)$是D的邻接矩阵。让$丁\mathrm{电子}G\left(丁\right)$是D的顶点出度的对角矩阵。对于任何实$\alpha \in [0,\mathrm{1个}]$, 广义邻接矩阵${\mathrm{一个}}_{\alpha }\left(丁\right)$有向图D的定义为${\mathrm{一个}}_{\alpha }\left(丁\right)=\alpha 丁\mathrm{电子}G\left(丁\right)+(\mathrm{1个}-\alpha )\mathrm{一个}\left(丁\right).$的特征值的最大模数${\mathrm{一个}}_{\alpha }\left(丁\right)$称为广义邻接谱半径或${\mathrm{一个}}_{\alpha }$-D的光谱半径。在本文中，我们获得了光谱半径的一些下界${\mathrm{一个}}_{\alpha }\left(丁\right)$根据有向图D的顶点数、弧数和闭合步数。确定达到这些下限的极值图。我们还获得了光谱半径的一些界限${\mathrm{一个}}_{\alpha }\left(丁\right)$在顶点出度方面，D和参数α的顶点的顶点平均2出度。我们描述了达到这些界限的极值有向图。