Enumeration of (3, 6)-Fullerenes

TL;DR

This paper provides an exact enumeration method for (3,6)-fullerenes, a class of cubic planar graphs with faces of 3 or 6 sides, including symmetric variants, based on prime factorization and quadratic equations.

Contribution

It introduces a novel enumeration approach for (3,6)-fullerenes using prime factorization and modular quadratic equations, extending previous combinatorial enumeration techniques.

Findings

Exact counts for (3,6)-fullerenes with any number of vertices.

Enumeration of symmetric (3,6)-fullerenes with specific symmetries.

Counts expressed via prime factorization and solutions to quadratic equations modulo primes.

Abstract

A (3, 6)-fullerene is a cubic planar graph whose faces all have 3 or 6 sides. We give an exact count of the number of (3, 6)-fullerenes for any given number of vertices. We also enumerate (3,6)-fullerenes with mirror symmetry, with 3-fold rotational symmetry, and with both types of symmetry. The counts are given in terms of the prime factorization of the number of vertices, by considering solutions to the quadratic equation $x^2 + x + 1 = 0$ modulo the primes in this prime factorization.