The size of switching classes with skew gains

Gain graphs are graphs in which each edge has a gain (a label from a group, say Γ, so that reversing the direction inverts the gain). In this paper we take a generalized view of gain graphs in which the gain of an edge is related to the gain of the reverse edge by an anti-involution. These gain graphs will be called skew gain graphs. We define switching by a selector, a generalization of switching (or Seidel switching) of an undirected graph. In this paper we compute the sizes of the resulting equivalence classes of skew gain graphs. This size can be determined by computing the size of an appropriate subgroup of Γ. We first examine the case that the graph is complete. Then we show how to reduce the general problem to connected graphs and prove that if the graph is connected, but not bipartite, it can be reduced to the complete case. The connected, bipartite case is solved separately.