The generalized X-join of Cayley graphs

TL;DR

This paper establishes conditions under which the generalized X-join of Cayley graphs results in a Cayley graph, expanding understanding of graph construction and symmetry properties in algebraic graph theory.

Contribution

It introduces new properties for a generalized wreath product of permutation groups and applies these to characterize when generalized X-joins of Cayley graphs are Cayley graphs.

Findings

Generalized X-join of isomorphic Cayley graphs is always vertex transitive.

Conditions are provided for the generalized wreath product to contain a regular subgroup.

The paper characterizes when the generalized X-join of Cayley graphs results in a Cayley graph.

Abstract

As a main result of this paper we give conditions under which the generalized $X$-join of Cayley graphs is a Cayley graph. In particular, we show that $X$-join of isomorphic Cayley graphs is a Cayley grpah. To do this, new properties for a generalized wreath product of permutation groups are given in the case where the base group acts regularly. These are used to give conditions for the generalized wreath product to contain a regular subgroup, which are then applied to generalized $X$-joins of Cayley graphs. Along the way, it is shown that the generalized $X$-join of isomorphic Cayley graphs will always be a vertex transitive graph.