两类六角系统的$A\alpha$-特征多项式和$A\alpha$-谱

On the $A_{\alpha}$-Characteristic Polynomials and the $A_{\alpha}$-Spectra of Two Classes of Hexagonal Systems

中文关键词:  $A\alpha$-特征多项式  $A\alpha$-谱  六角系统

英文关键词:$A_{\alpha}$-characteristic polynomial  $A_{\alpha}$-spectrum  hexagonal system

中文摘要:

      2017年, Nikiforov首次提出研究图$G$的$A\alpha$-矩阵, 其定义为:$A\alpha(G)=\alpha D(G)+(1-\alpha)A(G) (\alpha\in [0,1])$, 其中$A(G)$和$D(G)$分别为图$G$的邻接矩阵和度对角矩阵. 设$F_n$和$M_n$分别为圈状六角系统和M\"{o}bius带状六角系统图. 根据循环矩阵的行列式和特征值, 本文首先给出图$F_n$和$M_n$的$A\alph$-特征多项式和$A\alpha$-谱, 进一步得到图$F_n$和$M_n$的$A\alpha$-能量的上界.

英文摘要:

      The $A_{\alpha}$-matrix of a graph $G$ is defined as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ $(\alpha\in[0,1])$, given by Nikiforov in 2017, where $A(G)$ and $D(G)$ are, respectively, the adjacency matrix and the degree matrix of graph $G$. Let $F_{n}$ and $M_{n}$ be hexacyclic system graph and M\"{o}bius hexacyclic system graph, respectively. In this paper, according to the determinant and the eigenvalues of a circulant matrix, we firstly present $A_{\alpha}$-characteristic polynomial and $A_{\alpha}$-spectrum of $F_{n}$ (resp., $M_{n}$). Furthermore, we obtain the upper bound of the $A_{\alpha}$-energy of $F_{n}$ (resp., $M_{n}$).