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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.