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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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. $M_{n}$).