Abstract
Let G be a connected d-regular graph and \(\lambda _2(G)\) be the second largest eigenvalue of its adjacency matrix. Mohar and O (private communication) asked a challenging problem: what is the best upper bound for \(\lambda _2(G)\) which guarantees that \(\kappa (G) \ge t+1\), where \(1 \le t \le d-1\) and \(\kappa (G)\) is the vertex-connectivity of G, which was also mentioned by Cioabă. As a starting point, we determine a sharp bound for \(\lambda _2(G)\) to guarantee \(\kappa (G) \ge 2\) (i.e., the case that \(t =1\) in this problem), and characterize all families of extremal graphs.
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Acknowledgements
We thank the two anonymous referees for their valuable comments and constructive suggestions, which helped us improve the quality of our work.
Funding
Jianfeng Wang was supported by the National Natural Science Foundation of China (Grant number 11971274) and Special Fund for Taishan Scholars Project. Wenqian Zhang was supported by the Tianyuan Fund for Mathematics, National Natural Science Foundation of China (Grant number 12226304) and Natural Science Foundation of Shandong Province (Grant number ZR2022QA045).
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Zhang, W., Wang, J. Sharp spectral bounds for the vertex-connectivity of regular graphs. J Comb Optim 45, 65 (2023). https://doi.org/10.1007/s10878-023-00992-0
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DOI: https://doi.org/10.1007/s10878-023-00992-0