On real number labelings and graph invertibility

Abstract

For non-negative real x0 and simple graph G, λx0,1(G) is the minimum span over all labelings that assign real numbers to the vertices of G such that adjacent vertices receive labels that differ by at least x0 and vertices at distance two receive labels that differ by at least 1. In this paper, we introduce the concept of λ-invertibility: G is λ-invertible if and only if for all positive x, λx,1(G)=xλ1x,1(Gc). We explore the conditions under which a graph is λ-invertible, and apply the results to the calculation of the function λx,1(G) for certain λ-invertible graphs G. We give families of λ-invertible graphs, including certain Kneser graphs, line graphs of complete multipartite graphs, and self-complementary graphs. We also derive the complete list of all λ-invertible graphs with maximum degree 3.