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

Abstract:

      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., $M_{n}$).