Enumerating the Classes of Local Equivalency in Graphs

There are local operators on (labeled) graphs G with labels (gij) coming from a finite field. If the filed is binary (in other words, if the graph is ordinary), the operations are just the local complementations, but in the general case, there are two different types of operators. The first type is the following: let v be a vertex of the graph and a ∈ Fq, and we obtain a graph with labels g ′ ij = gij + agvigvj . For the second type of operators, let 0 6= b ∈ Fq and the resulted graph is a graph with labels g ′′ vi = bgvi and g ij = gij , for i, j unequal to v. In the present article, we show that the number of graphs locally equivalent to a graph is at most q, and the number of classes of local equivalency is q n 2 2 .