A study of the Structural Properties of finite $G$-graphs and their Characterisation

TL;DR

This paper explores the structural properties and characterizations of finite and infinite G-graphs derived from groups, analyzing their spectra, energy, and specific examples like symmetric and semi-dihedral groups.

Contribution

It provides a comprehensive characterization of finite G-graphs, explores their dependence on generating sets, and computes spectral properties and energies for various classes of G-graphs.

Findings

Characterization of finite G-graphs in general and bipartite cases.

Spectral analysis and energy computation for various G-graphs.

Examples of G-graphs for symmetric, alternating, and semi-dihedral groups.

Abstract

The $G$-graph $\Gamma(G,S)$ is a graph from the group $G$ generated by $S\subseteq G$, where the vertices are the right cosets of the cyclic subgroups $\langle s \rangle, s\in S$ with $k$-edges between two distinct cosets if there is an intersection of $k$ elements. In this thesis, after presenting some important properties of $G$-graphs, we show how the $G$-graph depends on the generating set of the group. We give the $G$-graphs of the symmetric group, alternating group and the semi-dihedral group with respect to various generating sets. We give a characterisation of finite $G$-graphs; in the general case and a bipartite case. Using these characterisations, we give several classes of graphs that are $G$-graphs. For instance, we consider the Tur\'{a}n graphs, the platonic graphs and biregular graphs such as the Levi graphs of geometric configurations. We emphasis the structural properties of $G$-graphs and their relations to the group $G$ and the generating set $S$. As preliminary results for further studies, we give the adjacency matrix and spectrum of various finite $G$-graphs. As an application, we compute the energy of these graphs. We also present some preliminary results on infinite $G$-graphs where we consider the $G$-graphs of the infinite group $SL_2(\mathbb{Z})$ and an infinite non-Abelian matrix group.