Graphs with Minimal Strength

Abstract

For any graph G of order p, a bijection f:V(G)→{1,2,…,p} is called a numbering of G. The strength strf(G) of a numbering f of G is defined by strf(G)=max{f(u)+f(v)|uv∈E(G)}, and the strength str(G) of a graph G is str(G)=min{strf(G)|f is a numbering of G}. In this paper, many open problems are solved, and the strengths of new families of graphs are determined.