We prove that the underlying graphs of p-gonal r-valent orientably regular maps are arc-transitive but non-Cayley if r 3 and p is a prime greater than r(r ? 1).

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 l...