On Cayley Graphs on the Symmetric Group Generated by Tranpositions

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, we denote by its smallest non-zero Laplacian eigenvalue. In this paper we show that among all sets of n-1 transpositions which generate the symmetric group, , the set whose associated Cayley graph has the highest is the set {(1, n), (2, n), ..., (n-1, n)} (or the same with n and i exchanged for any i<n). For this set we have . This result follows easily from the following result. For any set of transpositions, T, we can form a graph on n vertices, , by forming an edge {i, j} in for each transposition . We prove that if is bipartite, then of the Cayley graph associated to T is at most ; left open is the compelling conjecture that the two 's are always equal. We discuss this and other generalizations of Bacher's work, which dealt with the case T = {(1, 2), (2, 3), ..., (n-1, n)}.