Main Article Content
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.