Derangement action digraphs and graphs

TL;DR

This paper introduces derangement action digraphs, a new class of graphs based on derangements, explores their structural and symmetry properties, and identifies conditions under which they form simple graphs, with potential applications in network modeling.

Contribution

The paper defines derangement action digraphs, characterizes when they are simple graphs, and analyzes their structural and symmetry properties, expanding the understanding of group action graphs.

Findings

Conditions for derangement action digraphs to be simple graphs

Structural properties of derangement action digraphs

Symmetry and automorphism characteristics

Abstract

We study the family of \emph{derangement action digraphs}, which are a subfamily of the group action graphs introduced in [Fred Annexstein, Marc Baumslag, and Arnold L. Rosenberg, Group action graphs and parallel architectures, \emph{SIAM J. Comput.} 19 (1990), no. 3, 544--569]. For any non-empty set $X$ and a non-empty subset $S$ of $\Der(X)$, the set of derangments of $X$, we define the derangement action digraph $\rm\overrightarrow{DA}(X;S)$ to have vertex set $X$, and an arc from $x$ to $y$ if and only if $y=x^s$ for some $s\in S$. In common with Cayley graphs and digraphs, derangement action digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on $S$ under which $\rm\overrightarrow{DA}(X;S)$ may be viewed as a simple graph of valency $|S|$, and we call such graphs derangement action graphs. Also we investigate the structural and symmetry properties of these digraphs and graphs. Several open problems are posed and many examples are given.