Asymptotic enumeration of perfect matchings in $m$-barrel fullerene graphs

TL;DR

This paper asymptotically counts the number of perfect matchings in $m$-barrel fullerene graphs with many hexagonal layers using two different methods, confirming the consistency of the results.

Contribution

It introduces two novel methods for asymptotic enumeration of perfect matchings in $m$-barrel fullerene graphs and verifies their agreement.

Findings

Both methods yield identical asymptotic counts.

The number of perfect matchings grows exponentially with the number of layers.

The results provide insights into the combinatorial structure of fullerene graphs.

Abstract

A connected planar cubic graph is called an $m$-barrel fullerene and denoted by $F(m,k)$, if it has the following structure: The first circle is an $m$-gon. Then $m$-gon is bounded by $m$ pentagons. After that we have additional k layers of hexagons. At the last circle $m$-pentagons connected to the second $m$-gon. In this paper we asymptotically count by two different methods the number of perfect matchings in $m$-barrel fullerene graphs, as the number of hexagonal layers is large, and show that the results are equal.