Enumeration of finite field labels on graphs

Abstract

Let G be a graph, let R be a finite field and Γ a subgroup of the automorphism group Aut G of G. For a labeling of the vertices of G with elements of R, we consider labelings of the edges of G with elements of R such that the label on each edge is equal to the sum of labels on the two incident vertices. Such an edge labeling is called induced from some vertex labeling. We enumerate the number of Γ-isomorphism classes of induced edge labelings for G and R.