G-graphs Characterisation and Incidence Graphs

TL;DR

This paper provides an extended characterization of G-graphs, explores their relationship with incidence graphs, and investigates which complete graphs can be represented as incidence graphs of G-graphs.

Contribution

It offers a comprehensive characterization of G-graphs and establishes a link between bipartite G-graphs and incidence graphs, revealing significant overlaps.

Findings

Extended characterization of G-graphs

Established connection between bipartite G-graphs and incidence graphs

Partial results on incidence graphs of complete graphs

Abstract

Graphs derived from groups are a widely studied class of graphs, motivated by their highly symmetric structure. In particular, G-graphs offer an easy and interesting alternative construction of semi-symmetric graphs. After recalling the main properties of these graphs, this papers gives an extended characterisation of G-graphs and develops the link between bipartite G-graphs and incidence graphs. It appears that these two classes of graphs have a wide overlapping despite having completely different constructions. We give partial answers to the problem of finding which complete simple graphs have a G-graph as their incidence graph.