An improved Moore bound for mixed graphs

TL;DR

This paper derives a tighter Moore-like bound for the maximum size of mixed graphs with diameter at least three and fully classifies optimal (1,1)-regular mixed graphs with diameter three, confirming the bound's optimality.

Contribution

It introduces an improved Moore bound for mixed graphs and provides a complete enumeration of optimal (1,1)-regular graphs with diameter three.

Findings

Derived an improved Moore-like bound for mixed graphs

Enumerated all optimal (1,1)-regular mixed graphs with diameter three

Proved the bound's optimality through complete classification

Abstract

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.