#### Title

#### Date of Award

12-2018

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Ping Zhang

#### Second Advisor

Dr. Gary Chartrand

#### Third Advisor

Dr. Allen Schwenk

#### Fourth Advisor

Dr. Garry Johns

#### Keywords

Generalized line graph, graph structure, derived graph

#### Abstract

With every nonempty graph, there are associated many graphs. One of the best known and most studied of these is the line graph *L (G)* of a graph *G*, whose vertices are the edges of *G* and where two vertices of *L (G)* are adjacent if the corresponding edges of *G* are adjacent. This concept was implicitly introduced by Whitney in 1932. Over the years, characterizations of graphs that are line graphs have been given, as well as graphs whose line graphs have some specified property. For example, Beineke characterized graphs that are line graphs by forbidding certain graphs that can be subgroups. Sedlacek characterized those graphs whose line graph is planar. Harary and Nash-Williams characterized those graphs whose line graph is Hamiltonian. Chartrand and Wall proved that if G is a connected graph all of whose vertices have degree 3 or more, then, although * L(G)* may not be Hamiltonian, the line graph of * L(G)* must be Hamiltonian.

Over the years, various generalizations of line graphs have been introduced and studied by many. Among them are Schwenk graphs and *k*-line graphs introduced in 2015 and 2016 here at Western Michigan University. This study introduces a generalization of line graphs and discusses several well-known structural properties of this class of graphs. Furthermore, it establishes a number of characterizations of connected graphs whose generalized line graphs possess some prescribed graph structure.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Alqahtani, Mohra Abdullah Z., "Generalized Line Graphs" (2018). *Dissertations*. 3348.

https://scholarworks.wmich.edu/dissertations/3348