# Some results on double triangle descendants of $K_5$

## TL;DR

This paper investigates a special class of graphs generated from $K_5$ via double triangle expansion, providing enumeration results and properties relevant to quantum field theory and graph invariants.

## Contribution

It introduces and analyzes double triangle descendants of $K_5$, including enumeration with generating functions and minimal triangle counts, linking graph theory to physics.

## Key findings

- Explicit rational generating functions for certain descendants
- Minimum of four triangles in any descendant
- Enumeration of graphs with up to four additional vertices

## 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.