## Theses and Dissertations

#### Issuing Body

Mississippi State University

#### Advisor

Dobson, Ted

#### Committee Member

McBride, Matthew

#### Committee Member

Fabel, Paul

#### Committee Member

Sivaraman, Vaidyanathan

#### Committee Member

Smith, Robert; Woody, Jon

#### Date of Degree

4-30-2021

#### Original embargo terms

Worldwide

#### Document Type

Dissertation - Open Access

#### Major

Mathematics

#### Degree Name

Doctor of Philosophy

#### College

College of Arts and Sciences

#### Department

Department of Mathematics and Statistics

#### Abstract

It is known that a Cayley digraph of an abelian group A is isomorphic to a nontrivial wreath product if and only if there is a proper nontrivial subgroup B of A such that the connection set without B is a union of cosets of B in A. We generalize this result to Cayley digraphs of nonabelian groups G by showing that such a digraph is isomorphic to a nontrivial wreath product if and only if there is a proper nontrivial subgroup H of G such that S without H is a union of double cosets of H in G. This result is proven in the more general situation of a double coset digraph (also known as a Sabidussi coset digraph.) We then give applications of this result which include obtaining a graph theoretic definition of double coset digraphs, and determining the relationship between a double coset digraph and its corresponding Cayley digraph. We further expand the result obtained for double coset digraphs to a collection of bipartite graphs called bi-coset graphs and the bipartite equivalent to Cayley graphs called Haar graphs. Instead of considering when this collection of graphs is a wreath product, we consider the more general graph product known as an X-join by showing that a connected bi-coset graph of a group G with respect to some subgroups L and R of G is isomorphic to an X-join of a collection of empty graphs if and only if the connection set is a union of double cosets of some subgroups N containing L and M containing R in G. The automorphism group of such -joins is also found. We also prove that disconnected bi-coset graphs are always isomorphic to a wreath product of an empty graph with a bi-coset graph.

#### Recommended Citation

Barber, Rachel V., "Recognizing algebraically constructed graphs which are wreath products." (2021). *Theses and Dissertations*. 5080.

https://scholarsjunction.msstate.edu/td/5080