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 $\delta$ of vertices in each of the other independent sets. We determine when a Moore multipartite digraph is weakly distance-regular. Within this framework, some necessary conditions for the existence of a Moore $r$-partite digraph with interpartite outdegree $\delta>1$ and diameter $k=2m$ are obtained. In the case $\delta=1$, 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

On the Spectrum of a Weakly Distance-Regular Digraph

The notion of distance-regularity for undirected graphs can be extended for the directed case in two different ways. Damerell adopted the strongest definition of distanceregularity, which is equivalent to say that the corresponding set of distance matrices {Ai} D i=0 constitutes a commutative association scheme. In particular, a (strongly) distance-regular digraph Γ is stable, which means that A ⊤ i = Ag−i, for each i = 1,...,g − 1, where g denotes the girth of Γ. If we remove the stability property from the definition of distance-regularity, it still holds that the number of walks of a given length between any two vertices of Γ does not depend on the chosen vertices but only on their distance. We consider the class of digraphs characterized by such a weaker condition, referred to as weakly distance-regular digraphs, and show that their spectrum can also be obtained from a smaller ‘quotient digraph’. As happens in the case of distance-regular graphs, the study is greatly facilitated by a family of orthogonal polynomials called the distance polynomials.