THE NUMBER OF SMALL CYCLES IN THE STAR GRAPH

The Star graph is the Cayley graph on the symmetric group Sym(n) generated by the set of transpositions { (1 i) is an element of Sym(n) : 2 <= i 6 <= g. 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 (pi, i d) - cycles the cycles constructed from two shortest paths between a given vertex pi and the identity i d. In this paper we derive the exact number of (pi; i d) cycles for particular structures of the vertex pi. We use these results to obtain the total number of 10-cycles passing through any given vertex in the Star graph.