On the Strength and Independence Number of Graphs
Rikio Ichishima, Francesc Antoni Muntaner-Batle, Yukio Takahashi · Contributions to Mathematics · 2022
A numbering f of a graph G of order n is a labeling that assigns distinct elements of the set {1, 2, . . ., n} to the vertices of G. The strength strIn this paper, we present a necessary and sufficient condition for the strength of a graph G of order n to meet the constraints str (G) = 2n -2β (G) + 1 and str (G) = n + δ (G) = 2n -2β (G) + 1, where β (G) and δ (G) denote the independence number and the minimum degree of G, respectively.This answers open problems posed by Gao, Lau, and Shiu [Symmetry 13 (2021) #513].Also, an earlier result leads us to determine a formula for the strength of graphs containing a particular class of graphs as a subgraph.We also extend what is known in the literature about k-stable properties.