On the nonexistence of almost Moore digraphs with self-repeats

TL;DR

This paper investigates the structure of almost Moore digraphs with self-repeats, proving their nonexistence for certain degrees and diameters, thereby advancing the understanding of their possible configurations and supporting the conjecture of their overall nonexistence.

Contribution

It extends nonexistence results of almost Moore digraphs with self-repeats to larger diameters and degrees, providing new conditions under which these digraphs cannot exist.

Findings

Proves nonexistence of almost Moore digraphs with self-repeats for degrees 4 and 5 when diameter is large enough.

Establishes nonexistence for degrees 6 to 12 with diameter greater than 2.

Provides structural insights into automorphisms and subdigraphs of these digraphs.

Abstract

An almost Moore digraph is a diregular digraph of degree $d>1$, diameter $k>1$ and order $d+d^2+ \cdots +d^k$. Their existence has only been shown for $k=2$. It has also been conjectured that there are no more almost Moore digraphs, but so far their nonexistence has only been proven for $k=3,4$ and for $d=2,3$ when $k\geq 3$. In this paper we study the structure of the subdigraphs of an almost Moore digraph induced by the vertices fixed by an automorphism determined by a power of the permutation $r$ of repeats of the digraph. We deduce that each almost Moore digraph of degree $d$ and diameter $k$ with self-repeats has such a subdigraph whose vertices have order $\leq d-1$ under $r$. From this, we extend the results about the nonexistence of almost Moore digraphs with self-repeats of degrees 4 and 5 to those whose diameter is large enough with respect to the degree. More precisely, we prove their nonexistence when $k\geq 2(d-1)$ if $k$ is odd and when $k \geq 2(d-1)^2$ if $k$ is even. We also show that these findings jointly with other results imply that there are no almost Moore digraphs with self-repeats for degrees $d$, $6\leq d\leq 12$, and $k>2$.