Mixed Moore Cayley Graphs

The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. There has been much recent interest in the problem for mixed graphs, where we allow both undirected edges and directed arcs in the graph. For a diameter 2 graph with maximum undirected degree r and directed out-degree z, a straightforward counting argument yields an upper bound M(z,r,2)=(z+r)2+z+1 for the order of the graph. Apart from the case r=1, the only three known examples of mixed graphs attaining this bound are Cayley graphs, and there are an infinite number of feasible pairs (r,z) where the existence of mixed Moore graphs with these parameters is unknown. We use a combination of elementary group-theoretical arguments and computational techniques to rule out the existence of further examples of mixed Cayley graphs attaining the Moore bound for all orders up to 485

Decompositions of complete bipartite and tripartite graphs into selfcomplementary factors with finite diameters

We completely determine the spectrum of the complete bipartite and tripartite graphs that are decomposable into two isomorphic factors with a finite diameter

Multipartite Moore digraphs

We derive some Moore-like bounds for multipartite digraphs, which extend those of bipartite digraphs, under the assumption that every vertex of a given partite set is adjacent to the same number δ of vertices in each of the other independent sets. We determine when a multipartite Moore digraph is weakly distance-regular. Within this framework, some necessary conditions for the existence of a r-partite Moore digraph with interpartite outdegree δ > 1 and diameter k = 2m are obtained. In the case δ = 1, which corresponds to almost Moore digraphs, a necessary condition in terms of the permutation cycle structure is derived. Additionally, we present some constructions of dense multipartite digraphs of diameter two that are vertex-transitive