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Presentations for vertex-transitive graphs.

Agelos Georgakopoulos¹, Alex Wendland¹

¹Department of Mathematics, University of Warwick, Coventry, CV4 7AL UK.

Journal of Algebraic Combinatorics

|May 10, 2022

Summary

This summary is machine-generated.

This study introduces a generalized group presentation method to construct Cayley graphs. This new approach enables the representation of all vertex-transitive graphs, expanding graph theory possibilities.

Keywords:

Bi-Cayley graph Cayley graph Group presentation Vertex-transitive

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Area of Science:

Graph Theory
Group Theory
Combinatorics

Background:

Standard Cayley graph constructions rely on uniform group presentations.
Vertex-transitive graphs are a fundamental class of graphs with broad applications.
Existing methods may not encompass all vertex-transitive graph structures.

Purpose of the Study:

To generalize standard Cayley graph constructions.
To develop a novel notion of group presentation.
To demonstrate the representation of every vertex-transitive graph.

Main Methods:

Generalizing group presentations by allowing varied relators for different vertices.
Applying these generalized presentations to graph construction.
Proving the encompassment of all vertex-transitive graphs.

Main Results:

A new definition of group presentation is established.
This generalized presentation allows for the construction of any vertex-transitive graph.
The method provides a unified framework for representing diverse graph structures.

Conclusions:

The generalized group presentation offers a powerful tool for graph theory.
This work expands the scope of Cayley graph constructions.
It provides a new perspective on understanding vertex-transitive graphs.