The number of small cycles in the star graph

The Star graph is the Cayley graph on the symmetric group Symn generated by the set of transpositions ((1 i) ε Sym_n: 2 ≤ i ≤ n). This graph is bipartite and does not contain odd cycles but contains all even cycles with a sole exception of 4-cycles. We denote as (π, id)-cycles the cycles constructed from two shortest paths between a given vertex π and the identity id. In this paper we derive the exact number of (π, id)- cycles for particular structures of the vertex π. We use these results to obtain the total number of 10-cycles passing through any given vertex in the Star graph.