Almost Moore and the largest mixed graphs of diameters two and three

TL;DR

This paper characterizes almost Moore mixed graphs for diameters 2 and 3, providing a complete understanding of their structure and some optimal constructions for other diameters, advancing the degree/diameter problem.

Contribution

It offers a complete characterization of almost Moore mixed graphs for diameters 2 and 3, and proposes optimal constructions for other diameters, filling a gap in the degree/diameter problem.

Findings

Complete characterization for diameters 2 and 3

New constructions for other diameters

Advances in understanding extremal mixed graphs

Abstract

Almost Moore mixed 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 unit less than the Moore bound for such graphs. The problem of their existence has been considered just for diameter $2$. In this paper, we give a complete characterization of these extremal mixed graphs for diameters 2 and 3. We also derive some optimal constructions for other diameters.