Abstract
Denote by \(\rho (G)\) and \(\kappa (G)\) the spectral radius and the signless Laplacian spectral radius of a graph G, respectively. Let \(k\ge 0\) be a fixed integer and G be a graph of size m which is large enough. We show that if \(\rho (G)\ge \sqrt{m-k}\), then \(C_4\subseteq G\) or \(K_{1,m-k}\subseteq G\). Moreover, we prove that if \(\kappa (G)\ge m-k+1\), then \(K_{1,m-k}\subseteq G\). Both these results extend some known results.
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We would like to gratefully thank anonymous referees for their careful reading and valuable comments which led to an improved version of the paper.
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This work is supported by National Natural Science Foundation of China (Nos. 12171154, 12301438, 1211101361), the China Postdoctoral Science Foundation (No. 2021M691671) and Chenguang Program of Shanghai Education Development Foundation and Shanghai Municipal Education Commission.
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Wang, Z., Guo, JM. Maximum degree and spectral radius of graphs in terms of size. J Algebr Comb 59, 213–224 (2024). https://doi.org/10.1007/s10801-023-01289-5
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DOI: https://doi.org/10.1007/s10801-023-01289-5