## Mathematics, Department of

#### Date of this Version

8-2015

#### Abstract

The *center* of a graph is the set of vertices whose distance to other vertices is minimal. The *centralizing* *number* of a graph *G* is the minimum number of additional vertices in any graph* H* where *V*(*G*) is the center of *H*. Buckley, Miller, and Slater and He and Liu provided infinite families of graphs with each centralizing number. We show the number of graphs with each nonzero centralizing number grows super-exponentially with the number of vertices. We also provide a method of altering graphs without changing the centralizing number and give results about the centralizing number of dense graphs.

The *degree sequence* of a (hyper)graph is the list of the number of edges containing each vertex. A *t-switch* replaces* t* edges with* t* new edges while maintaining the same degree sequence. For graphs, it has been repeatedly shown that any realization of a degree sequence can be turned into any other realization by a sequence of 2-switches. However, Gabelman provided an example to show 2-switches are not sufficient for *k*-graphs with *k* ≥ 3. We classify all pairs of 3-graphs that do not admit a 2-switch but differ by a 3-switch. We use this to provide support that 2-switches and a 3-switch are sufficient for 3-graphs.

Given graphs *G* and *H*, *G* is *H-saturated* if *G* does not contain *H* as a subgraph, but *H* is a subgraph of *G+e* for any *e* not in *E*(*G*). While this is well defined for subgraphs, the similar definition is not well defined for induced subgraphs. To avoid this, Martin and Smith defined the* induced-saturation number* using trigraphs. We show that the induced-saturation number of stars is zero. This implies the existence of graphs that are star induced-saturated. We introduce the parameter indsat*(*n,H*) which is the minimum number of edges in an *H*-induced-saturated graph, when one exists. We provide bounds for indsat*(*n,K*_{1,3}) and compute it exactly for infinitely many *n*.

Adviser: Stephen Hartke

## Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Stephen Hartke. Lincoln, Nebraska: August, 2015

Copyright (c) 2015 Sarah Lynne Behrens