Vertex-connectivity and eigenvalues of graphs

Abstract

Let κ(G), μn−1(G), λ2(G) and q2(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 μn−1(G)>H2(Δ,δ,k) or λ2(G)<δ−H2(Δ,δ,k) or q2(G)<2δ−H2(Δ,δ,k), where H2(Δ,δ,k)=(k−1)nΔ(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 μn−1(G), λ2(G) and q2(G) to characterize vertex-connectivity of regular graphs, triangle-free graphs and graphs with fixed girth are also presented.