Enumerating the Classes of Local Equivalency in Graphs

There are local operators on (labeled) graphs $G$ with labels $(g_{ij})$ coming from a finite field. If the filed is binary, in other words, if the graph is ordinary, the operation is just the local complementation. That is, to choose a vertex and complement the subgraph induced by its neighbors. 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 $a\in \mathbf{F}_q$, the finite field of $q$ elements. The operator is to obtain a graph with labels $g'_{ij}=g_{ij}+ag_{vi}g_{vj}$. For the second type of operators, let $0\neq b\in \mathbf{F}_q$ and the resulted graph is a graph with labels $g''_{vi}=bg_{vi}$ and $g''_{ij}=g_{ij}$, for $i,j$ unequal to $v$. The local complementation operator (binary case) has appeared in combinatorial theory, and its properties have studied in the literature. Recently, a profound relation between local operators on graphs and quantum stabilizer codes has been found, and it has become a natural question to recognize equivalency classes under these operators. In the present article, we show that the number of graphs locally equivalent to a given graph is at most $q^{2n+1}$, and consequently, the number of classes of local equivalency is $q^{\frac{n^2}{2}-o(n)}$.