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.2021CXZX-594).
Author NameAffiliation
Mengyue YUAN Institute of Applied Mathematics, Lanzhou Jiaotong University, Gansu 730070, P. R. China 
Fei WEN Institute of Applied Mathematics, Lanzhou Jiaotong University, Gansu 730070, P. R. China 
Ranran WANG Institute of Applied Mathematics, Lanzhou Jiaotong University, Gansu 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, 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:2095-2651.2023.03.002
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