I have really had two very distinct research careers. The first started with myPh.D. research on finite simple groups in 1970, and continued for the first few years of my career at Allegheny College. During this time I published a few papers on finite simple groups and solvability of factorizable groups. In 1979 I recieved an NSF Grant to learn about some area of applied math. I chose applied graph theory and combinatorial matrix theory, and spent a year in Boulder working with John Maybee. This was the start of a long and fruitful collaboration. The original motivation of our work was to use graph theory to simplify the study of large matrix models of energy distribution systems. We then recognized that our approach could be used to study competition graphs which have applications to a problem in ecology. The work on competition graphs led naturally to applications related to the conflict graph of a communication network and frequency assignment problems. One of the ideas used in much of our work on competition graphs involved edge clique covers. This led us to considering directed biclique covers of digraphs which led to a considerable body of work in combinatorial matrix theory.

Applied Graph Theory:
The main thrust of my research from 1980-2000 was on competition graphs and their generalizations such as p-competition graphs, p-neighborhood graphs, and tolerance competition graphs. Work in this area also led to several papers on interval graphs, chordal graphs and niche graphs, an area of research we introduced in 1987. In the mid 90's we considered the competition graph of a tournament. This led us to a related graph, the domination graph of a tournament. Two vertices x and y dominate a tournament if for every other vertex z, either x beats z or y beats z. Two such vertices are called a dominant pair. The domination graph of a tournament T, denoted dom(T), is a graph on the vertices of T with edges between vertices which are dominant pairs. Over the last several years in a long series of papers we have completely characterized domination graphs of all tournaments as well as domination graphs of some special classes such as regular tournaments. Generalizing this work, we have also characterized mixed pair graphs of tournaments, as well as getting results on near-regular tournaments and domination-compliance graphs, work which remains in progress. More recently we obtained several results on domination graphs of complete paired comparison digraphs. Some of the work on domination graphs also led to considering quadrangular tournaments, and work on this project continues. Since 2000, along with several graduate students, I have been working on generalizations of interval graphs called interval bigraphs, interval k-graphs, and probe interval graphs. We have several results in this area when we have restricted the problem either to the unit case or some particular types of bipartite graphs, but there remain several interesting open questions. Most recently we have characterized interval tournaments.

Combinatorial Matrix Theory:
While using edge clique covers to characterize competition graphs, we were led to consider covering bipartite graphs with bicliques and digraphs with directed bicliques. For each of these problems we considered the minimum biclique covering and partition numbers. We then discovered that this was related to two matrix ranks of the corresponding adjacency matrices, namely, the boolean rank and nonnegative integer rank. In fact, we can also consider the normal real rank as well as the term rank of the matrices. So there are nice relationships between two graph parameters and four matrix ranks. There are many interesting and unsolved problems in this general area. One of them is to find the minimum boolean rank of an n x n tournament matrix, which is related to an unsolved problem on tournament codes. Most recently we have come back to the following major problem. For what {0,1}-matrices are some or all of these ranks equal? We have results on some classes, namely, nearly reducible matrics, nearly decomposable matrices, unipathic digraphs, dominoe free bipartite graphs, and some classes of tournaments, but in general the problem remains wide open and our work continues.

Research Colleagues:
Over the past twenty years I have been fortunate to work with so many wonderful colleagues from across the country and around the world as well as several Ph.D. students. Of course, I owe my introduction to this research area to John Maybee, with whom I published 27 papers between 1981 and 1997 when he died unexpectedly. Unfortunately, another of my long time colleagues, Norm Pullman, passed away a few years ago. I was fortunate to have worked with each of these distinguished mathematicians. Other coauthors include the following: Brooks Reid, Cal State San Marcos; Steve Bowser and Chuck Cable, Allegheny College; Dave Fisher, Kathy Fraughnaugh, and Harvey Greenberg, Univ. of Colorado at Denver; Fred Roberts, Rutgers University; Frank Harary, New Mexico State Univ.; Buck McMorris, Univ. of Louisville; Terry McKee, Wright State Univ.; Dom de Caen, David Gregory, and Rolf Rees, Kingston University; Curt Barefoot, New Mexico Tech; Suzanne Seager, Mount St. Vincent University; Terrie Henson, Naval Postgraduate School; Larry Langley, Univ. of Pacific; Han Cho, Seoul National University; and Suh Kim, Kyung Hee University.

I have also done joint work with all of my former Ph.D. students including the following: Kim Factor, Lindenwood University; Craig Rasmussen, Naval Postgraduate School; Charlie Anderson, Jeffco School District and Univ. of Colorado at Denver; Sarah Merz, Univ. of Pacific; Patty McKenna, Metro State Univ.; Guillermo Jimenez, Raytheon Advanced Technologies; Faun Doherty, Univ. of Colorado at Denver; Daluss Siewert, S. Dakota St. Univ.; Dave Brown, Utah State Univ; Dustin Stewart, Trinity University.