Distinguishing graphs by the number of homomorphisms
A homomorphism from one graph to another is a map that sends vertices to vertices and edges to edges. We denote the number of homomorphisms from G to H by |G → H|. If 𝓕 is a collection of graphs, we say that 𝓕 distinguishes graphs G and H if there is some member X of 𝓕 such that |G → X | ≠ |H → X|. 𝓕 is a distinguishing family if it distinguishes all pairs of graphs. We show that various collections of graphs are a distinguishing family.