On λ_1-extremal non-regular graphs

Abstract

Let G be a connected non-regular graph with n vertices, maximum degree ∆ and
minimum degree δ, and let λ₁ be the greatest eigenvalue of the adjacency matrix of G. In this paper, by studying the Perron vector of G, it is shown that type-I-a graphs and type-I-b (resp. type-II-a) graphs with some specified properties are not λ₁-extremal graphs. Moreover, for each connected non-regular graph some lower bounds on the difference between ∆ and λ₁ are obtained.