  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
$q^{\frac{n^2}{2}-o(n)}$.
