On the $A_{\alpha}$Characteristic Polynomials and the $A_{\alpha}$Spectra of Two Classes of Hexagonal Systems 
Received:April 28, 2022 Revised:August 22, 2022 
Key Words:
$A_{\alpha}$characteristic polynomial $A_{\alpha}$spectrum hexagonal system

Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11961041), the Natural Science Foundation of Gansu Province (Grant No.21JR11RA065) and the Excellent Postgraduates of Gansu Provincial Department of Education ``Star of Innovation'' Foundation (Grant No.2021CXZX594). 

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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., $M_{n}$). 
Citation: 
DOI:10.3770/j.issn:20952651.2023.03.002 
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