The spectral radius of graphs with given independence number

Abstract

As a classic problem of spectral graph theory, Brualdi-Solheid problem asks which graph achieves the extremal (maximum or minimum) spectral radius for a given class of graphs. In this paper, we focus on this problem for graphs with fixed independence number. We first obtain the maximum spectral radius and unique extremal graph among all bipartite graphs of order n with independence number α. Secondly, we study the minimum spectral radius of graphs with fixed independence number. Let Gn,α be the set of connected graphs of order n with independence number α. We show that tree has the minimum spectral radius over all graphs in Gn,α provided that α≥⌈n2⌉, and determine all the extremal graphs in Gn,n−4.