On Mixed Almost Moore Graphs of Diameter Two

On Mixed Almost Moore Graphs of Diameter Two

Mixed almost Moore graphs appear in the context of the Degree/Diameter problem as a class of extremal mixed graphs, in the sense that their order is one less than the Moore bound for mixed graphs. The problem of their existence has been considered before for directed graphs and undirected ones, but not for the mixed case, which is a kind of generalization. In this paper we give some necessary c...

A Generalization of Moore Graphs of Diameter Two

On the nonexistence of almost Moore digraphs of diameter four

Almost Moore digraphs appear in the context of the degree/diameter problem as a class of extremal directed graphs, in the sense that their order is one less than the unattainable Moore bound M(d, k) = 1 + d + · · · + d, where d > 1 and k > 1 denote the maximum out-degree and diameter, respectively. So far, the problem of their existence has only been solved when d = 2, 3 or k = 2, 3. In this pa...

Approaching the mixed Moore bound for diameter two by Cayley graphs

In a mixed (Δ, d)-regular graph, every vertex is incident with Δ ≥ 1 undirected edges and there are d ≥ 1 directed edges entering and leaving each vertex. If such a mixed graph has diameter 2, then its order cannot exceed (Δ+ d) + d+1. This quantity generalizes the Moore bounds for diameter 2 in the case of undirected graphs (when d = 0) and digraphs (when Δ = 0). For every d such that d − 1 is...

Moore Graphs of Diameter 2 and 3

In [1], the authors prove that for k = 2, there exists unique Moore graphs of type (2, 2), (3, 2) and (7, 2). Moreover, they show that only other possible (d, 2) graph is (57, 2), but could not confirm its existence. For k = 3, the authors show that the only Moore graph is (2, 3) and it is unique. Amazingly, the existence of the (57, 2) graph remains an open problem to this day. To prove these ...