The A α-spectral radius of dense graphs

Abstract

A c-cyclic graph is a connected graph with n vertices and n + c−1 edges. In this paper, we consider the problem that: in the class of graphs, each of which is the complement graph of a c-cyclic graph with n vertices, which graph has the largest -spectral radius. We shall show that, for , the extremal graph must be a graph with an isolated vertex. This implies that the maximum degree of G is larger, then the -spectral radius of its complement graph is also larger. We also determine the structure of the extremal graph with largest -spectral radius among the class of graphs, each of which is the complement graph of a c-cyclic graph with n vertices and maximum degree n−1 for and n−2 for , respectively.