Graphs encoding the generating properties of a finite group

TL;DR

This paper introduces a family of graphs associated with finite groups, where vertices are tuples of group elements and edges indicate generating sets, to explore how these graphs reflect the underlying group's structure.

Contribution

It defines and analyzes the properties of the graphs $\Gamma_{a,b}(G)$, revealing how their features encode information about the finite group $G$.

Findings

Analysis of isolated vertices and loops in the graphs

Connections between graph connectivity and group generation properties

Insights into how graph diameter relates to group structure

Abstract

Assume that $G$ is a finite group. For every $a, b \in\mathbb N,$ we define a graph $\Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and $(y_1,\dots,y_b)$ are adjacent if and only if $\langle x_1,\dots,x_a,y_1,\dots,y_b \rangle =G.$ We study several properties of these graphs (isolated vertices, loops, connectivity, diameter of the connected components) and we investigate the relations between their properties and the group structure, with the aim of understanding which information about $G$ are encoded by these graphs.