Connectivity and eigenvalues of graphs with given girth or clique number

Abstract

Let κ ′ (G), κ(G), µ n−1 (G) and µ 1 (G) denote the edge-connectivity, vertex-connectivity, the algebraic connectivity and the Laplacian spectral radius of G, respectively. In this paper, we prove that for integers k ≥ 2 and r ≥ 2, and any simple graph G of order n with minimum degree δ ≥ k, girth g ≥ 3 and clique number ω(G) ≤ r, the edge-connectivity, where N (δ, g) is the Moore bound on the smallest possible number of vertices such that there exists a δ-regular simple graph with girth g, and ϕ(δ, r) = max{δ + 1, ⌊ rδ r−1 ⌋}. Analogue results involving µ n−1 (G) and µ 1 (G) µ n−1 (G) to characterize vertex-connectivity of graphs with fixed girth and clique number are also presented. Former results in [