Bicoset groups and symmetric graphs

In this paper we construct a functor from the set of bicoset multivalued groups with one Hermite generator to the set of symmetric graphs. The functor makes it possible to desribe the set of bicoset multivalued groups with one Hermite generator as a sum of categories. The categories are indexed by a pair $(\Gamma^U, G^U)$ where $G^U$ is a universal symmetric graph and $G^U$ is a subgroup of $\operatorname{Aut}\Gamma^U$.