Vertex-connectivity and eigenvalues of graphs

Let kappa(G), mu(n-1)(G) lambda(2)(G) and q(2)(G) denote the vertex-connectivity, the algebraic connectivity, the second largest adjacency eigenvalue, and the second largest signless Laplacian eigenvalue of G, respectively. In this paper, we prove that for an integer k > 0 and any simple graph G of order n with maximum degree Delta and minimum degree delta >= k, the vertex-connectivity kappa(G) >= k if mu(n-1)(G) > H-2(Delta, delta, k) or lambda(2)(G) < delta - H-2(Delta, delta, k) or q(2)(G) < 2 delta - H-2(Delta, delta, k), where H-2(Delta, delta, k) = (k-1)n Delta/(n-k+1)(k-1)+4(delta-k+2) (n-delta-1), which improves the result in [Appl. Math. Comput. 344-345 (2019) 141-149] and the result in [Electron. J. Linear Algebra 34 (2018) 428-443]. Analogue results involving mu(n-1)(G), lambda(2)(G) and q(2)(G) to characterize vertex-connectivity of regular graphs, triangle-free graphs and graphs with fixed girth are also presented. (C) 2019 Elsevier Inc. All rights reserved.