On the $A_{\alpha}$Characteristic Polynomials and the $A_{\alpha}$spectra of Two Classes of Hexagonal Systems 
Received:April 28, 2022 Revised:August 14, 2022 
Key Words:
$A_{\alpha}$characteristic polynomial $A_{\alpha}$spectrum hexagonal system

Fund Project:National Natural Science Foundation of China (Grant No.11961041) 
Author Name  Affiliation  Address  Mengyue Yuan  Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China  Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China  Fei Wen  Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China  Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China  Ranran Wang  Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China  Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China 

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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, in accordance with the determinant and characteristic eigenvalue of circulant matrix, we firstly present the $A_{\alpha}$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. $M_{n}$). 
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