MOD SUM NUMBER OF COMPLETE BIPARTITE GRAPHS

A Thesis
Presented to the Faculty of the Department of Mathematics
East Tennessee State University






In Partial Fulfillment
of the Requirements for the Degree
Master of Science in Mathematical Sciences






by
Christopher D. Wallace
May, 1999

APPROVAL
This is to certify that the Graduate Committee of
Christopher D. Wallace
met on the
1st day of April, 1999.

The committee read and examined his thesis, supervised his defense of it in an oral examination, and decided to recommend that his study be submitted to the Graduate Council, in partial fulfillment of the requirements for the degree of Master of Science in Mathematics.



James W. Boland
Chair, Graduate Committee


Robert B. Gardner


Debra J. Knisley


Linda M. Lawson


Signed on behalf of
the Graduate Council Dr. Brown, Dean
School of Graduate Studies

ABSTRACT
MOD SUM NUMBER OF COMPLETE BIPARTITE GRAPHS
by

Christopher D. Wallace

A graph $G=(V,E)$ is a mod sum graph if there exists a positive integer $\cal Z$ and a labelling of vertices with distinct elements of $\{1,2,...,{\cal{Z}}-1\}$ such that $\{u,v\} \in E$ if and only if $u \ne v$ and $u+v$ $(mod$ $\cal Z)$ $\in V$. First we discuss conditions which $K_{n_1,n_2,...,n_m}$ must satisfy to be a mod sum graph and then we determine the minimum number of isolated vertices such that $K_{n,m}$ $m \ge n \ge 3$ is a mod sum graph except when $2n \le m < 3n-3$ and $m$ is odd. pt

Copyright by Christopher D. Wallace 1999

DEDICATION

To my wife Olivia, my parents Billie and Bill, and my sister Sheena. Their support, encouragement, and love made this thesis possible.

ACKNOWLEDGEMENTS

I would like to thank Jay Boland who helped with assistance in making this thesis a reality.