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Published in:
01-03-2023
Sharp spectral bounds for the vertex-connectivity of regular graphs
Published in: Journal of Combinatorial Optimization | Issue 2/2023
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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.