Date of Award
Doctor of Philosophy
Topological graph theorists study the imbeddings of graphs on surfaces (spheres with handles). Some interesting questions in the field are on w hat surfaces can a graph be 2 -cell imbedded and how m any such imbeddings are there on each surface. The study of these and related questions is called Enumerative Topological Graph Theory. Random Topological Graph Theory uses probability models to study the 2-cell imbeddings. It generalizes the results from Enumerative Topological Graph Theory (which is the uniform case, p= 1/2) to an arbitrary probability p.
We study the model where the sample space consists of all labeled, orientable 2-cell imbeddings of a fixed connected cubic graph. In this model, neighbors of a vertex are either oriented clockwise (with probability p, 0 < p < 1) or counter-clockwise (with probability 1-p). By giving a global definition of “clockwise”, we construct probability polynomials for each surface upon which a fixed graph can be imbedded and for the expected value of the genus random variable. We investigate properties of these probability polynomials and give probability polynomials for a number of small order cubic graphs as well as some families of graphs.
Tesar, Esther Joy, "Probability Polynomials for Cubic Graphs in the Framework of Random Topological Graph Theory" (1995). Dissertations. 1767.