The Q-minimizer graph with given independence number

Abstract

Let Gn,α be the set of all connected graphs of order n with independence number α. A graph is called the Q-minimizer graph (A-minimizer graph) if it attains the minimum signless Laplacian spectral radius (adjacency spectral radius) over all graphs in Gn,α. In this paper, we first show that the Q-minimizer graph must be a tree for α≥⌈n2⌉, and then we derive seven propositions about the Q-minimizer graph. Moreover, when n−α is a constant, the structure of the Q-minimizer graph is characterized. The method of getting Q-minimizer graph in this paper is different from that of getting A-minimizer graph. As applications, we determine the Q-minimizer graphs for α=n−1,n−2,n−3 and n−4, respectively. The results of α=n−1,n−2,n−3 are consistent with that in Li and Shu (2010) [15] and the result of α=n−4 is new. Interestingly, the Q-minimizer graph in Gn,n−4 is unique, which is exactly one of the A-minimizer graphs in Gn,n−4.