Vertex-connectivity and eigenvalues of graphs.

Let κ (G) , μ n − 1 (G) , λ 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 Δ and minimum degree δ ≥ k , the vertex-connectivity κ (G) ≥ k if μ n − 1 (G) > H 2 (Δ , δ , k) or λ 2 (G) < δ − H 2 (Δ , δ , k) or q 2 (G) < 2 δ − H 2 (Δ , δ , k) , where H 2 (Δ , δ , k) = (k − 1) n Δ (n − k + 1) (k − 1) + 4 (δ − k + 2) (n − δ − 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 μ n − 1 (G) , λ 2 (G) and q 2 (G) to characterize vertex-connectivity of regular graphs, triangle-free graphs and graphs with fixed girth are also presented. [ABSTRACT FROM AUTHOR]