The extreme eigenvalues and maximum degree of k-connected irregular graphs

Let λ1(G)λ1(G) be the largest eigenvalue and λn(G)λn(G) be the smallest eigenvalue of a k-connected irregular graph G with n vertices, m edges and maximum degree Δ. In this paper, we prove thatΔ−λ1(G)>(nΔ−2m)k2(nΔ−2m)[n2−2(n−k)]+nk2. This inequality improves previous results of several authors and implies two lower bounds on Δ+λn(G)Δ+λn(G) which also refine some known bounds. Another lower bound on Δ−λ1(G)Δ−λ1(G) for a connected irregular graph G is given as well.