#### Date of Award

12-2017

#### 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

#### Abstract

One of the most familiar derived graphs are line graphs. The line graph *L(G)* of a graph *G* is the graph whose vertices are the edges of* G* where two vertices of *L(G)* are adjacent if and only if the corresponding edges of* G* are adjacent. One of the best- known results on the structure of line graphs deals with forbidden subgraphs by Beineke. A characterization of graphs whose line graph is Hamiltonian is due to Harary and Nash-Williams. Iterated line graphs of almost all connected graphs were shown to be Hamiltonian by Chartrand. The girth of a graph *G* is the length of a smallest cycle of* G*. An *r*-regular graph of girth *g* of minimum order is called a cage. Another class of derived graphs having a connection with cages was introduced by Schwenk. For a graph *G* having girth 2*k* + 1, the Schwenk graph *G** of *G* has the set of all (*k* + 1)-paths as its vertex set where two vertices *P* and *Q* are adjacent in *G** if and only if* P* and *Q* have only an end-vertex in common and the vertices of* P* and *Q* induce a (2*k* + 1)-cycle. In this work, we introduce two new classes of derived graphs, called ℓ -line graphs and* Z*-graphs. The concept of ℓ -line graphs is a generalization of line graphs and Schwenk graphs, while the *Z*-graphs provide a different view of certain line graphs. We primarily study the structures of these derived graphs. Results, conjectures and problems on the structural properties such that connectedness, decompositions, Hamiltonicity and planarity of these graphs are presented.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Alhulwah, Khawlah Hamad, "Structures of Derived Graphs" (2017). *Dissertations*. 3204.

https://scholarworks.wmich.edu/dissertations/3204