#### Date of Award

8-1994

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Allen Schwenk

#### Second Advisor

Yousef Alavi

#### Third Advisor

Gary Chartrand

#### Fourth Advisor

Arthur T. White

#### Abstract

The expander coefficient of a graph is a parameter that is utilized to quantify the rate at which information is spread throughout a graph. The eigenvalues of the Lapladan of a graph provide a bound for the expander coefficient of the graph. In this dissertation, we construct many pairs of isospectral graphs with different expander coefficients.

In Chapter I, we define the problem and present some preliminary definitions. We then introduce two constructions that are related to graph composition and that may be employed to produce cospectral and isospectral graphs.

In Chapter II, we investigate the connectivity of and distance in the graphs formed by the constructions of Chapter I.

In Chapter HI, we construct four pairs of infinite sequences of isospectral graphs. For each pair of infinite sequences, it is demonstrated that the expander coefficients of the corresponding graphs of the sequences are different. We determine the limit of each sequence of each pair, and see that these limits need not be equal.

In Chapter IV, the Folkman graph is used to construct a pair of isospectral graphs that more accurately model networks. The expander coefficients for the graphs are then shown to be unequal.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Walters, Ian Campbell Jr., "Isospectral Graphs and the Expander Coefficient" (1994). *Dissertations*. 1849.

http://scholarworks.wmich.edu/dissertations/1849

## Comments

Fifth Advisor: Dr. Joseph Buckley

Sixth Advisor: Dr. Morteza Shafii-Mousavi