Relative Cayley graphs of finite groups

TL;DR

This paper investigates the properties of relative Cayley graphs of finite groups, focusing on their connectivity, forbidden structures, and numerical invariants, to deepen understanding of their algebraic and combinatorial characteristics.

Contribution

It introduces the concept of relative Cayley graphs with respect to subgroups and analyzes their structural properties and invariants, expanding the theory of Cayley graphs.

Findings

Analysis of connectivity conditions

Identification of forbidden substructures

Calculation of key numerical invariants

Abstract

The relative Cayley graph of a group $G$ with respect to its proper subgroup $H$, is a graph whose vertices are elements of $G$ and two vertices $h\in H$ and $g\in G$ are adjacent if $g=hc$ for some $c\in C$, where $C$ is an inversed-closed subset of $G$. We study the relative Cayley graphs and, among other results, we discuss on their connectivity and forbidden structures, and compute some of their important numerical invariants.