Double Triangle Descendants of K5

Feynman diagrams in phi^4 theory can be represented as 4-regular graphs. The Feynman integral, or even the Feynman period, is very hard to calculate. A graph invariant, called the c2-invariant, is conjecturally thought to be equal for two graphs when their periods are equal. Double triangle reduction of 4-regular graphs is known to preserve the c2-invariant. Double triangle descendants of K5 all have a c2-invariant that is a constant -1, and conjecturally, are the only graphs with this c2-invariant. This thesis studies the structure of K5-descendants to gain insight on the c2-invariant, get closer to solving the conjecture, and to study what is an interesting combinatorial operation in its own right. It will be shown that the minimum number of triangles in a descendant is 4. Closed-form generating functions are found for three families of K5-descendants. Two encodings, one for n-zigzags, and a general one for all K5-descendants, are found.