The signless Laplacian spectral radius of k-connected irregular graphs

Let G be a k-connected irregular graph with n vertices, m edges, maximum degree Δ and minimum degree δ. In this paper, we mainly show2Δ−q1(G)>2(nΔ−2m)k22(nΔ−2m)[(n−2+2k−Δ)(n−δ−1)+k2]+nk2, where q1(G) is the signless Laplacian spectral radius of G. The inequality improves previous bounds of several authors in some cases. It also implies a lower bound of 2Δ−q1(H) for a proper subgraph H of a k-connected Δ-regular graph. Another lower bound of 2Δ−q1(G) for a connected graph G is also given.