关于自环图能量的新下界

New Lower Bounds on the Energy of Graphs with Self-Loops

摘要: 设$G_S$ 为$n$ 个顶点的自环图，其通过在简单图$G$ 上向点集$S\subseteq V\left(G\right)$ 中的点添加自环得到。自环图$G_S$ 的能量由Gutman等学者定义为$E\left(G_S\right)={\displaystyle {\sum }_{i=\text{1}}^n\left|{\lambda }_i-\frac{\sigma }{n}\right|}$ ，其中${\lambda }_{\text{1}},\cdots ,{\lambda }_n$ 为$G_S$ 的邻接特征值，$\sigma$ 为$G_S$ 中自环的个数。本文利用矩阵理论给出了一个自环图最大特征值上界的充要条件，并利用这个上界和Ozeki不等式分别给出了自环图能量$E\left(G_S\right)$ 的新下界。

Abstract: Let $G_S$ be a self-loop graph with $n$ vertices obtained from a simple graph $G$ by attaching one self-loop at each vertex in $S\subseteq V\left(G\right)$ . The energy of a self-loop graph $G_S$ is defined by Gutman et al. as $E\left(G_S\right)={\displaystyle {\sum }_{i=\text{1}}^n\left|{\lambda }_i-\frac{\sigma }{n}\right|}$ , where ${\lambda }_{\text{1}},\cdots ,{\lambda }_n$ are the adjacency eigenvalues of $G_S$ and $\sigma$ is the number of self-loops of $G_S$ . In this paper, the sufficient and necessary conditions for the upper bound of the maximum eigenvalue of a self-loop graph are given. Moreover, new lower bounds on the energy of graphs with self-loops $E\left(G_S\right)$ are obtained by using this upper bound and Ozeki’s inequality, respectively.