#### Date of Award

4-2019

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Ping Zhang

#### Second Advisor

Dr. Gary Chartrand

#### Third Advisor

Dr. Clifton Ealy

#### Fourth Advisor

Dr. Allen Schwenk

#### Abstract

Perhaps the most fundamental property that a graph can possess is that of being connected. Two vertices u and v of a graph *G* are connected if *G* contains a *u-v* path. The graph *G* itself is connected if every two vertices of *G* are connected. The well-studied concept of connectivity provides a measure on how strongly connected a graph may be. There are many other degrees of connectedness for a graph. A Hamiltonian path in a graph *G* is a path containing every vertex of *G*. Among the best-known classes of highly connected graph are the Hamiltonian-connected graphs, in which every two vertices are connected by a Hamiltonian path. In many instances, graphs under study are required to have each pair of its vertices connected by a path with some prescribed property. For example, in a friendship graph, every pair of vertices is required to be connected by a unique path of length 2. In this work, we introduce the new concept of uniformly connected graphs which combines several features of connectedness of graphs in the literature. More precisely, for a positive integer k, a nontrivial connected graph *G* is k-uniformly connected if every two vertices of *G* are connected by a path of length k. The uniform spectrum of *G* is the set of all positive integers k for which *G* is k-uniformly connected. We study the structural properties of uniform spectra of connected graphs and investigate the existence of uniformly connected graphs possessing some prescribed properties.

#### Access Setting

Dissertation-Campus Only

#### Restricted to Campus until

4-2021

#### Recommended Citation

Almohanna, Nasreen, "Uniformly Connected Graphs" (2019). *Dissertations*. 3428.

https://scholarworks.wmich.edu/dissertations/3428