On free products of graphs

TL;DR

This paper introduces a new definition of free products of graphs that maximizes automorphism groups and explores their properties, including criteria for non-discreteness and automorphism classifications.

Contribution

It proposes a novel free product construction for graphs, extending existing definitions and analyzing automorphism groups and their classifications.

Findings

The new free product aligns with existing definitions for vertex-transitive graphs.

Criteria for the automorphism group to be non-discrete are established.

Tits' classification and simplicity criteria are adapted to free products of graphs.

Abstract

We define a free product of connected simple graphs that is equivalent to several existing definitions when the graphs are vertex-transitive but differs otherwise. The new definition is designed for the automorphism group of the free product to be as large as possible, and we give sufficient criteria for it to be non-discrete. Finally, we transfer Tits' classification of automorphisms of trees and simplicity criterion to free products of graphs.