Discrete Mathematics

Discrete mathematics embraces number theory, graph theory, experimental designs, cryptography and coding theory. These topics are called 'discrete' to distinguish them from the 'continuous' mathematics of Calculus and Analysis. Number theory is among the oldest and most traditional branches of mathematics but has important modern applications. The other topics are new developments underpinning the vast applications of information technology and communication in modern life.

Researchers:

Diophantine Equations in Geometry and Algebra
(Jim MacDougall)

We are interested in problems in geometry or algebra which reduce to solving diophantine equations. One such example is the search for polygons and polyhedra with integer-length sides whose area or volume is an integer. Another is the attempt to characterize rational polynomials in one variable which have rational roots and whose derivatives also have rational roots. A third is the study of graphs whose adjacency matrices have integral eigenvalues. These problems frequently involve the study of elliptic curves of higher genus equations. Collaborators on this project are R. Buchholz (Canberra) and C. Chisholm (Wollongong).

Graph Labelling
(Jim MacDougall)

A classic problem in graph theory is to prove that the complete graph K_n can be decomposed into isomorphic copies of any tree of order n. This problem can be attacked by studying the so-called graceful labellings of trees. We are searching for new constructions for graceful and related labellings for large classes of trees.

A magic labeling of a graph is a generalization of the familiar notion of magic square in which the vertices and edges are labelled with the consecutive integers so that all vertices or all edges have the same sum. As well as searching for methods of constructing such labellings, we are endeavouring to find general properties of a graph which guarantee the existence of such labellings. Collaborators on this project are W. Wallis (USA) and Ian Gray (Australia).