Physical Sciences and Mathematics Commons^™

Transformations Of Graphs And Digraphs, Elzbieta B. Jarrett

Dissertations

Some distances defined on graphs depend on transforming one graph into another. Two of these transformations are edge rotation and edge slide. In this dissertation, extensions and generalizations of these transformations are investigated.

Chapter I begins with some preliminary definitions and known results. Then two types of digraph transformations are introduced and their properties are studied.

Some measures of distance between graphs and distance between digraphs are defined in Chapter II. Also distance graphs and digraphs associated with these measures are introduced. Several known results concerning this topic are generalized and new results are presented.

Chapter III is devoted to …

A Representation Of Chemical Reactions By Labeled Graphs, Héctor Hevia

Dissertations

Graphs can be used to represent the atomic structure of chemical compounds where the vertices of the graph represent the individual atoms and the edges of the graph represent the valence bonds between a pair of atoms. M. A. Johnson (1991) introduced a graph-theoretic way to represent structural changes in chemical compounds. Thus, certain labelings of graphs called transitional labelings can be thought as representing chemical equations. Associated with these labelings, we introduce a new invariant of a graph G called the transitional value of G. The transitional value of a graph G gives an indication of how dramatic a …

Surgical Techniques For Constructing Minimal Orientable Imbeddings Of Joins And Compositions Of Graphs, David L. Craft

Dissertations

The various cases within the proof of the Heawood Map-Coloring Theorem, which established the genus of the complete graphs, utilized various techniques--some for the first time. This activity spurred interest in determining the genus of various other classes of graphs. However, very few generally applicable techniques have been developed, beyond those used in the proof of this famous theorem. Finding genera of arbitrary graphs remains a very difficult problem.

In this dissertation, we describe two surgical techniques for imbedding graphs. The first construction, called a graphical surface,views an orientable surface as a fattened graph, i.e., vertices become spheres and edges …