Minimum Number of Triangles of K5 Descendants

Abstract

In the study of Quantum Field Theory and Feynman Periods, the operation of double triangle expansion plays an important role. This is largely due to double triangle expansions not affecting the maximum weight of the period. In this thesis, we take a look at the effects of double triangle expansions on K5 graphs. More specifically, given any graph G that can be obtained through a sequence of double triangle expansions on K5, we calculate the minimum number of triangles of any graph that can be obtained through double triangle expansions on G. While the minimum number of triangles of graphs that are obtained through double triangle expansions on K5 is already known, this is a generalization of that. This is done by understanding the structure of graphs that come from K5 and double triangle expansions, and how double triangle expansions relate to this structure. Commonly arising graphs are studied, and showed to be building blocks for more complicated graphs.