Probability Theory of Completely Labelled Random Multigraphs

Abstract

The roots of graph theory are obscure. The famous eighteenth-century Swiss mathematician Leonard Euler was perhaps the first to solve a problem using graphs when he was asked to consider the problem of the Königsberg bridges (in the 1730s). Problems in (finite) graph theory are often enumeration problems, and thus can become rather intricate to be solved. However, in the late 1950s and early 1960s the Hungarian mathematicians Paul Erdös and Alfred Rényi founded the theory of random graphs and used probabilistic methods (limit theorems) to by-pass enumeration problems. These problems also became secondary with the emergence of powerful computers. Thus, perhaps no topic in mathematics has enjoyed such explosive growth in recent years as graph theory. This stepchild of combinatorics and topology has emerged as a fascinating topic for research in its own right. Moreover, during the last two decades, calculus of graph theory has proved to be a valuable tool in applied mathematics and life sciences as well. Using graph-theoretic concepts, scientists study properties of real systems by modelling and simulation. The aim of graph-theoretic investigations is, in fact, the simplest topological structure after that of isolated points: The structure of a graph is that of “points” or “vertices”, and “edges” or “lines”.