A map is a connected topological graph $\Gamma$ cellularly embedded in a surface. In this paper, applying Tutte's algebraic representation of map, new ideas for enumerating non-equivalent orientable or non-orientable maps of graph are presented. By determining automorphisms of maps of Cayley graph $\Gamma={ \rm Cay}(G:S)$ with ${ \rm Aut} \Gamma\cong G\times H$ on locally, orientable and non-orientable surfaces, formulae for the number of non-equivalent maps of $\Gamma$ on surfaces (orientable, non-orientable or locally orientable) are obtained . Meanwhile, using reseults on GRR graph for finite groups, we enumerate the non-equivalent maps of GRR graph of symmetric groups, groups generated by 3 involutions and abelian groups on orientable or non-orientable surfaces.
Categories: math.GM math.CO