A mixed graph G can contain both (undirected) edges and arcs (directed edges). Here we derive an improved Moore-like bound for the maximum number of vertices of a mixed graph with diameter at least three. Moreover, a complete enumeration of all optimal (1, 1)-regular mixed graphs with diameter three is presented, so proving that, in general, the proposed bound cannot be improved.
Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this article, we consider the case where such graphs are bipartite. As main results, we show that in this context the Moore‐like bound is attained in the case of diameter k=3, and that bipartite‐mixed graphs of diameter k≥4 do not exist.
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 conditions for the existence of mixed almost Moore graphs of diameter two derived from the factorization in $\mathbb{Q}[x]$ of their characteristic polynomial. In this context, we deal with the irreducibility of $\Phi_i(x^2+x-(r-1))$, where $\Phi_i(x)$ denotes the i-th cyclotomic polynomial.
For graphs with maximum degree d and diameter k , an upper bound on the number of vertices in the graphs is provided by the well-known Moore bound (denoted by M d ,k ). Graphs that achieve this bound (Moore graphs) are very rare, and determining how close one can come to the Moore bound has been a major topic in graph theory. Of particular note in this regard are the cage problem and the degree/diameter problem. In this article, we take a different approach and consider questions that arise when we fix the number of vertices in the graph at the Moore bound, but relax, by one, the diameter constraint on a subset of the vertices. In this context, regular graphs of degree d , radius k , diameter k + 1, and order equal to M d ,k are called radially Moore graphs. We consider two specific questions. First, we consider the existence question (extending the work of Knor), and second, we consider some natural measures of how well a radially Moore graph approximates a Moore graph.
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