Graphs with the minimum spectral radius for given independence number

Abstract

Let G n,α be the set of connected graphs with order n and independence number α. Given k = n − α, the graph with minimum spectral radius among G n,α is called the minimizer graph. Stevanović in the classical book [D. Stevanović, Spectral Radius of Graphs, Academic Press, Amsterdam, 2015.] pointed that determining minimizer graph in G n,α appears to be a tough problem on page 96. Very recently, Lou and Guo in [15] proved that the minimizer graph of G n,α must be a tree if α ≥ ⌈ n 2 ⌉. In this paper, we further give the structural features for the minimizer graph in detail, and then provide of a constructing theorem for it. Thus, theoretically we completely determine the minimizer graphs in G n,α along with their spectral radius for any given k = n − α ≤ n 2 . As an application, we determine all the minimizer graphs in G n,α for α = n − 1, n − 2, n − 3, n − 4, n − 5, n − 6 along with their spectral radii, the first four results are known in [15,20] and the last two are new.