Presentations for vertex-transitive graphs

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.