THE MINIMUM ORDER OF A CAYLEY GRAPH WITH GIVEN DEGREE AND DIAMETER

In a previous paper, we proved the following: The order of a vertex-transitive directed graph with outdegree r and diameter p is at least 2 + 2r + (rho - 3)(1 + [r/2]). In the case of a Cayley graph, where r greater-than-or-equal-to 3 and rho greater-than-or-equal-to 5, we show here that the minimal order is (rho + 1)(1 + [r/2]). This bound improves the above one. In the Abelian case with rho greater-than-or-equal-to 6, we show that equality holds if and only if the graph is either the lexicographic product of a directed cycle with a complete graph or obtained from this last graph by deleting one 1-factor with a special structure.