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 \(A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)\), where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G respectively. The largest eigenvalue of \(A_{\alpha }(G)\) is called the \(A_{\alpha }\)-spectral radius of G. In this paper, we give sharp upper bounds on the \(A_{\alpha }\)-spectral radius of \(C_4\)-free graphs and Halin graphs for \(\alpha \in [1/2, 1)\) respectively.
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The authors are grateful to the anonymous referees for valuable suggestions and corrections which result in an improvement of the original manuscript.
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Shu-Guang Guo was supported by the National Natural Science Foundation of China (Nos. 12071411, 12171222).
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Guo, SG., Zhang, R. The Sharp Upper Bounds on the \(A_{\alpha }\)-Spectral Radius of \(C_4\)-Free Graphs and Halin Graphs. Graphs and Combinatorics 38, 19 (2022). https://doi.org/10.1007/s00373-021-02429-z
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DOI: https://doi.org/10.1007/s00373-021-02429-z