Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the nonexistence of almost Moore digraphs with self-repeats

Arnau Messegué, Josep Maria Miret

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 $k>2$.

On the nonexistence of almost Moore digraphs with self-repeats

Abstract

An almost Moore digraph is a diregular digraph of degree , diameter and order . Their existence has only been shown for . It has also been conjectured that there are no more almost Moore digraphs, but so far their nonexistence has only been proven for and for when . 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 of repeats of the digraph. We deduce that each almost Moore digraph of degree and diameter with self-repeats has such a subdigraph whose vertices have order under . 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 if is odd and when if 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 , , and .

Paper Structure

This paper contains 11 sections, 19 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Almost Moore digraphs
  3. The cyclotomic conjecture
  4. Subdigraphs of a $(d,k)$-digraph induced by the fixed vertices of an automorphism
  5. The subdigraphs $H_{\alpha}$
  6. Regularity of $H_{\alpha}$
  7. The subdigraphs $H_{\alpha}$ different from $C_k$ are almost Moore digraphs
  8. Permutation cycle structure
  9. Matrix approach
  10. Computing some traces of $\hbox{\boldmath $P$}^j\hbox{\boldmath $A$}^{\ell}$
  11. Nonexistence of $(d,k)$-digraphs with self-repeats having large diameter $k$ with respect to the degree $d$

Key Result

Lemma 1

Let $G$ be a $(d,k)$-digraph and let $\varphi$ be an automorphism of $G$. If $u$, $v$ are two distinct fixed vertices by $\varphi$, then any walk of length at most $k$ connecting $u$ with $v$ is fixed by $\varphi^2$.

Figures (3)

  • Figure 1: The case $n_j(1)$ for all $j$.
  • Figure 3: The connection between $w$ and $v_1$.
  • Figure 4: Values of $d$ and $k$ for the nonexistence of $(d,k)$-digraphs.

Theorems & Definitions (41)

  • Conjecture 1
  • Remark 1
  • Lemma 1
  • proof
  • Definition 1
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • ...and 31 more