Some results on double triangle descendants of $K_5$

Abstract

Double triangle expansion is an operation on 4 -regular graphs with at least one triangle which replaces a triangle with two triangles in a particular way. We study the class of graphs which can be obtained by repeated double triangle expansion beginning with the complete graph K_5 . These are called double triangle descendants of K_5 . We enumerate, with explicit rational generating functions, those double triangle descendants of K_5 with at most four more vertices than triangles. We also prove that the minimum number of triangles in any K_5 descendant is four. Double triangle descendants are an important class of graphs because of conjectured properties of their Feynman periods when they are viewed as scalar Feynman diagrams, and also because of conjectured properties of their c_2 invariants, an arithmetic graph invariant with quantum field theoretical applications.