#### Date of Award

4-1995

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Allen Schwenk

#### Second Advisor

Dr. Yousef Alavi

#### Third Advisor

Dr. Joseph Buckley

#### Fourth Advisor

Dr. Don Lick

#### Abstract

Two well-known classes of graphs, strongly regular graphs and cages, have been studied extensively by many researchers for a long period of time. In this dissertation, we mainly deal with semi-strongly regular graphs, a class of graphs including all strongly regular graphs, and (r, g, t)-cages, a generalization of the usual cage concept.

Chapter I introduces the two new concepts: semi-strongly regular graphs and generalized (r, g, t)-cages, gives necessary conditions for the existence of semi-strongly regular graphs and some interesting properties regarding common neighbors of pairs of vertices, and shows connections between these two new concepts and the old ones as well as connections between the two new concepts themselves.

In Chapter II, we study the existence problem of clique-disjoint semistrongly regular graphs. We give lower bounds for the order of clique-disjoint semi-strongly regular graphs in Section 1. Then in Section 2 we prove that the necessary conditions given in Chapter I are also sufficient for a cliquedisjoint semi-strongly regular graph of order n to exist when n is relatively large. And in Section 3, we show that certain clique-disjoint semi-strongly regular graphs can not exist when their orders n are too close to the lower bound.

Chapter III is devoted to generalized (r, g, t)-cages. We prove that an (r, g, t)-cage always exists in Section 1. Then we study the lower bounds for the order of (r, g, t)-cages in Section 2 and list some known (r, g, t)-cages in the last section.

Maximal graphs without C4 on 31 vertices, which is the smallest unsettled case to a conjecture of Erdos, are studied in Chapter IV and some open questions are mentioned in Chapter V.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Fan, Cong, "Semi-Strongly Regular Graphs and Generalized Cages" (1995). *Dissertations*. 1764.

http://scholarworks.wmich.edu/dissertations/1764