A derived graph: triangle graph of a graph

Abstract

In this dissertation we study a derived graph called the triangle graph of a graph. We start by giving a brief background on derived graphs, followed by some basic definitions in graph theory which are relevant in this work. We then discuss some well known classes of graphs and some well known graph operations. Thereafter, we give a known formal definition of a line graph, followed by a few examples and some well known results on line graphs. This leads to the introduction of the main structure of this work, the triangle graph of a graph. We define the triangle graph of a graph and clarify the concept with examples. Then we establish a few properties of a triangle graph of a graph, followed by establishing triangle graphs of certain classes of graphs. Finally, we conclude the dissertation by discussing vertex-join of a graph and the relationship between the graph G, and the triangle graph of a vertex-join of G.