On mixed radial Moore graphs of diameter 3

TL;DR

This paper investigates mixed radial Moore graphs of diameter 3, classifies their extremal properties, and provides infinite families of such graphs, including an optimal one, through computational and theoretical methods.

Contribution

It introduces the concept of mixed radial Moore graphs, classifies their extremal properties, and constructs infinite families with proven existence and optimality.

Findings

Exhaustive computer search identified top-ranked graphs for specific parameters.

Two infinite families of mixed radial Moore graphs are constructed for diameter 3.

One of the families is proven to be optimal.

Abstract

Radial Moore graphs and digraphs are extremal graphs related to the Moore ones where the distance-preserving spanning tree is preserved for some vertices. This leads to classify them according to their proximity to being a Moore graph or digraph. In this paper we deal with mixed radial Moore graphs, where the mixed setting allows edges and arcs as different elements. An exhaustive computer search shows the top ranked graphs for an specific set of parameters. Moreover, we study the problem of their existence by providing two infinite families for different values of the degrees and diameter $3$. One of these families turns out to be optimal.