A revised Moore bound for mixed graphs

TL;DR

This paper refines the Moore bound for mixed graphs, providing a more accurate upper limit on the number of vertices for graphs with given degree and diameter, applicable to all diameters and degree combinations.

Contribution

It introduces a corrected Moore bound for mixed graphs, improving upon previous bounds for diameters greater than three.

Findings

Derived a reduced upper bound for diameters > 3

Presented a corrected Moore bound valid for all diameters

Applicable to all combinations of undirected and directed degrees

Abstract

The degree-diameter problem seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter. It is known that graphs attaining the maximum possible value (the Moore bound) are extremely rare, but much activity is focused on finding new examples of graphs or families of graph with orders approaching the bound as closely as possible. There has been recent interest in this problem as it applies to mixed graphs, in which we allow some of the edges to be undirected and some directed. A 2008 paper of Nguyen and Miller derived an upper bound on the possible number of vertices of such graphs. We show that for diameters larger than three, this bound can be reduced and we present a corrected Moore bound for mixed graphs, valid for all diameters and for all combinations of undirected and directed degrees.