| Bounds on the $A_{\alpha}$-Spectral Radius of a $C_3$-Free Graph |
| Received:October 04, 2020 Revised:January 03, 2021 |
| Key Words:
$C_3$-free graph $k$-cycle graph $A_{\alpha}$-spectral radius bound
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| Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.12071411; 12171222). |
| Author Name | Affiliation | | Dongxia ZHU | School of Mathematics and Statistics, Qinghai Normal University, Qinghai 810008, P. R. China
School of Mathematics and Statistics, Yancheng Teachers University, Jiangsu 224002, P. R. China | | Shuguang GUO | School of Mathematics and Statistics, Yancheng Teachers University, Jiangsu 224002, P. R. China | | Rong ZHANG | School of Mathematics and Statistics, Yancheng Teachers University, Jiangsu 224002, P. R. China |
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| Abstract: |
| Let $G$ be a simple undirected graph. For any real number $\alpha \in[0,1]$, Nikiforov defined the $A_{\alpha}$-matrix of $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ in 2017, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of $G$, respectively. In this paper, we obtain a lower bound on the $A_{\alpha}$-spectral radius of a $C_3$-free graph for $\alpha \in[0, 1)$ and a sharp upper bound on the $A_{\alpha}$-spectral radius of a $C_3$-free $k$-cycle graph for $\alpha \in[1/2, 1)$. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2022.01.001 |
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