On real number labelings and graph invertibility

For non-negative real x0x0 and simple graph GG, λx0,1(G)λx0,1(G) is the minimum span over all labelings that assign real numbers to the vertices of GG such that adjacent vertices receive labels that differ by at least x0x0 and vertices at distance two receive labels that differ by at least 1. In this paper, we introduce the concept of λλ-invertibility: GG is λλ-invertible if and only if for all positive xx, λx,1(G)=xλ1x,1(Gc). We explore the conditions under which a graph is λλ-invertible, and apply the results to the calculation of the function λx,1(G)λx,1(G) for certain λλ-invertible graphs GG. We give families of λλ-invertible graphs, including certain Kneser graphs, line graphs of complete multipartite graphs, and self-complementary graphs. We also derive the complete list of all λλ-invertible graphs with maximum degree 3.