University of Colorado Denver College of Liberal Arts and Sciences

Bio

This fall is the start of my 30th year at UCD, and it has been a most rewarding career. When I came here in 1981, I was one of 3 new faculty charged with the responsibility of building a program in applied mathematics. At the time there was no modern major for undergraduates, and only 6 students in a small graduate program. We quickly revamped the undergraduate major, and then began recruiting graduate students. I became chair of the dept. and led the effort for a new PhD program which was approved in 1986. During the 80's the graduate program grew quickly to 80 students. Serving as chair was a highlight of my career, but during the 90's I turned my attention to my real passions of teaching and developing PhD students. Late in the 90's I recognized that while our PhD program had been successful, most of our students were local due to lack of funding. So I took over as chair of the graduate committee, and led an effort to improve our graduate program. This included increased financial support, reduced teaching loads for teaching assistants, and more extensive recruiting at the national and international level. At this time I started the campus wide PhD Committee, and we finally solved the funding issues for out-of-state students. As a consequence, we now have a much more diverse graduate program with many students from around the country and several foreign countries. I followed this effort with two more years as chair of the department . Now I am back again to teaching and working with PhD students as I slowly phase into retirement.
I have really had two very distinct research careers. The first started with my PhD research on finite simple groups in 1970 and continued for awhile during my early career at Allegheny College. Wanting to get into more applied research, I obtained an NSF grant in 1979, and spent a year working with John Maybee in Boulder. This led to a long and successful collaboration with John working in the areas of applied graph theory and combinatorial matrix theory, and also led to my position at UCD. The original thrust of this research was on competition graphs of digraphs and various generalizations such as niche graphs, and a number of interesting matrix rank problems associated with a variety of {0,1}-matrices, including tournament matrices. The combination of competition graphs and tournaments led to a long series of papers characterizing domination graphs of tournaments. For a long time I was intrigued with interval graphs, and for the last 10 years have worked on a variety of problems on variations of interval graphs such as interval bigraphs, interval p-graphs, and probe interval graphs. Most recently I have been involved in the characterization of interval tournaments and the characterization of k-trees that are interval p-graphs and probe interval graphs. Over the years I have published about 85 papers on these topics working with some 20 coauthors from around the world and 13 current and former PhD students. I think the proudest accomplishment with respect to research is the continued success of my PhD students in their own careers, either in academics or industry.

Select Publications

"A Class of Cycle-free Unit Probe Interval Graphs," Discrete Applied Mathematics, 157, 762-767(with D. Brown and L. Sheng), 2009

"Interval Tournaments," J. of Graph Theory DOI 10.1.1002/jgt, 72-81(with D. Brown and A. Busch), 2007

"Quadrangularity and Strong Quadrangularity in Tournaments," Australasian J. of Combinatorics 34, 247-260(with K.B. Reid, S. Severini, and D. Stewart), 2006

"Niche Graphs and Mixed Pair Graphs of Tournaments", J. of Graph Theory 31, 319-332 (with S. Bowser and C. Cable), 1999

"The Domination and Competition Graphs of Tournaments," J. of Graph Theory 29, 103-110(with D. Fisher, S. Merz, and K.B. Reid), 1998

"Competition Graphs of Strongly Connected and Hamiltonian Digraphs," SIAM J. on Discrete Math, Vol. 8, No. 2, 179-185(with K. Fraughnaugh. J. Maybee, S. Merz, and N. Pullman), 1995

"i,j-Competition Graphs," Discrete Applied Math 32, 241-262(with K. Hefner, S. Kim, K. Jones, and F. Roberts), 1991

"Biclique Coverings of Regular Digraphs and Minimum Semiring Ranks of Regular Matrices," J. of Combinatorial Theory, Series B, Vol 51, No. 11(with D. Gregory, K. Jones, and N. Pullman), 1991

"On Signed Digraphs with All Cycles Negative," Discrete Applied Math 12, 155-164(with F. Harary and J. Maybee), 1985

"On Finite Simple Groups All of Whose 2-Local Subgroups are Solvable," J. of Algebra 27, 491-515, 1973

Courses Taught

Math 3191: Applied Linear Algebra
Math 3410: Modern Algebra
Math 4408: Applied Graph Theory
Math 4793/5793: Discrete Models
Math 5718: Applied Linear Algebra
Math 6404: Graph Theory
Math 7413, 7414: Modern Algebra I, II
Math 7823: Topics in Discrete Math