Bounds on the $A_{\alpha}$-Spectral Radius of a $C_3$-Free Graph
Received:October 04, 2020  Revised:January 03, 2021
Key Word: $C_3$-free graph   $k$-cycle graph   $A_{\alpha}$-spectral radius   bound
Fund ProjectL: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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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 $\alpha \in[1/2, 1)$.