Bounds on the $A_{\alpha}$-spectral radius of a $C_3$-free graph
Received:October 04, 2020  Revised:October 04, 2020
Key Word: $C_3$-free graph   $k$-cycle graph   $A_{\alpha}$-spectral radius
Fund ProjectL:the National Natural Science Foundation of China (No. 12071411) and the Natural Science Foundation of the Jiangsu Higher Education Institutions.(No.18KJB110031).
 Author Name Affiliation Address Shu-Guang Guo Yancheng Teachers University 江苏省盐城市开放大道50号 盐城师范学院 数学与统计学院 Dong-Xia Zhu Yancheng Teachers University 江苏省盐城市开放大道50号 盐城师范学院 数学与统计学院 Rong Zhang Yancheng Teachers University 江苏省盐城市开放大道50号 盐城师范学院 数学与统计学院
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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 $\alpha \in[1/2, 1)$.