Connectivity and eigenvalues of graphs with given girth or clique number

Let kappa'(G), mu(n-1)(G) and mu(1)(G) denote the edge-connectivity, the algebraic connectivity and the Laplacian spectral radius of G, respectively. In this paper, we prove that for integers k >= 2and r >= 2, and any simple graph Gof order nwith minimum degree delta >= k, girth g >= 3and clique number.(G) = r, the edge-connectivity omega(G) <= r/mu(n-1)(G) =(k-1) nN(delta,g)(n-N(delta,g)) or if mu(n-1)(G) >= (k-1)/phi(delta,r)(n-phi(delta,r)), where N(delta, g) is the Moore bound on the smallest possible number of vertices such that there exists a delta-regular simple graph with girth g, and phi(delta, r) = max{delta+ 1, [r delta/r-1]. Analogue results involving mu(n-1)(G) and mu(1)(G) mu(n-1)(G) to characterize vertexconnectivity of graphs with fixed girth and clique number are also presented. Former results in Liu et al. (2013) [22], Liu et al. (2019) [20], Hong et al. (2019) [15], Liu et al. (2019) [21] and Abiad et al. (2018) [1] are improved or extended. (C) 2020 Elsevier Inc. All rights reserved.