#### Date of Award

6-1992

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Allen Schwenk

#### Second Advisor

Dr. Arthur T. White

#### Third Advisor

Dr. Paul Erdos

#### Fourth Advisor

Dr. Yousef Alavi

#### Abstract

Two sequences of nonnegative integers have the *k^th* common moment if they have equal sums of* k^th* powers. We intend to study common moment sets of the degree sequences of complementary graphs, and similarly, of the score sequences of complementary tournaments.

In Chapter I, we first study common moment sets of arbitrary sequences of nonnegative integers. Some basic concepts are introduced. The relations between characteristic functions and initial common moments are discovered. We extend Hua’s discussion of the Tarry-Escott problem. We conclude that any finite subset of nonnegative integers can be a common moment set of some sequences, and conversely, any common moment set must be finite. Complementary sequences are also discussed

We study common moments problems of the degree sequences of complementary graphs in Chapter II. Some interesting results on testing degree sequences are given. We explicitly construct complementary graphs which have the first *2p* + 1 moments in common for any *p*. Furthermore, we seek the smallest such graphs. This can be achieved with *cp^2* 1n *p* vertices. We introduce Pascal submatrices and study their singularity. Finally we characterize the sets which can be the common moment sets of complementary graphs.

In Chapter III, we study common moment sets of complementary tournaments. We give some new observations for testing score sequences. We present Schwenk’s construction of certain complementary tournaments. This provides us with a means of finding complementary tournaments of small order with desired initial common moments. We construct some examples of tournaments which possess a given common moment set. Finally, we characterize common moment sets using the concept of initial density.

Several open problems are presented in Chapter IV for further study.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Chen, Hang, "Common Moment Sets of Complementary Graphs" (1992). *Dissertations*. 1911.

https://scholarworks.wmich.edu/dissertations/1911

## Comments

Fifth Advisor: Dr. Joseph McCanna