Largest independent sets of certain regular subgraphs of the derangement graph

The derangement graph is the Cayley graph on the symmetric group (mathcal {S}_{n}) whose generating set (D_{n}) is the set of permutations on ([n]={1, ldots , n}) without any 1-cycle. For any fixed positive integer (k le n), the Cayley graph generated by the subset of (D_{n}) consisting of permutations without any i-cycles for all (1 le i le k) is a regular subgraph of the derangement graph. In this paper, we determine the smallest eigenvalue of these subgraphs and show that the set of all the largest independent sets in these subgraphs is equal to the set of all the largest independent sets in the derangement graph, provided n is sufficiently large in terms of k.