[Paper Review] Fullerenes with distant pentagons

Jan Goedgebeur, Brendan D. McKay|arXiv (Cornell University)|Aug 12, 2015

Fullerene Chemistry and Applications3 references3 citations

TL;DR

This paper identifies the smallest fullerenes with a given minimum distance between pentagons (pentagon separation d), proving that for each d > 0, the minimal fullerene is unique up to mirror image and is an icosahedral fullerene with Coxeter coordinates (⌈d/2⌉, ⌊d/2⌋) or (⌊d/2⌋, ⌈d/2⌋) for odd d, or (d/2, d/2) for even d. It further establishes that for each d, there exists a threshold h_d such that fullerenes with pentagon separation at least d exist for any number of hexagons ≥ h_d, and provides exact counts of such fullerenes up to 400 vertices for d = 1 to 5.

ABSTRACT

For each $d>0$, we find all the smallest fullerenes for which the least distance between two pentagons is $d$. We also show that for each $d$ there is an $h_d$ such that fullerenes with pentagons at least distance $d$ apart and any number of hexagons greater than or equal to $h_d$ exist. We also determine the number of fullerenes where the minimum distance between any two pentagons is at least $d$, for $1 \le d \le 5$, up to 400 vertices.

Motivation & Objective

• To determine the smallest fullerenes for which the minimum distance between any two pentagons is exactly d, for each d > 0.
• To prove that for each d, there exists a threshold h_d such that fullerenes with pentagon separation at least d exist for any number of hexagons ≥ h_d.
• To enumerate the number of fullerenes with pentagon separation at least d for 1 ≤ d ≤ 5, up to 400 vertices.
• To extend the Isolated Pentagon Rule (IPR) and the maximum pentagon separation rule by characterizing optimal pentagon distributions in fullerenes.
• To provide a complete computational census of fullerenes with increasing minimum pentagon separation up to 400 vertices.

Proposed method

• The authors use the dual graph representation of fullerenes, where pentagon separation corresponds to the distance between degree-5 vertices in a triangulation.
• They analyze the penta-hexagonal net to derive lower bounds on the number of vertices required for a given pentagon separation d.
• For odd d, they construct minimal fullerenes by assembling 12 disjoint patches of radius ⌊d/2⌋ around each pentagon, proving uniqueness via boundary compatibility constraints.
• For even d, they use a different patch type with radius d/2 − 1/2, leading to a unique minimal fullerene with Coxeter coordinates (d/2, d/2).
• They apply computational enumeration techniques to count fullerenes with pentagon separation at least d up to 400 vertices, using the Stevin Supercomputer Infrastructure.
• They validate results by comparing with known IPR fullerenes and extending prior work on IPR existence thresholds (h_1 and h_2).

Experimental results

Research questions

• RQ1 What is the smallest fullerene for which the minimum distance between any two pentagons is exactly d, for each d > 0?
• RQ2 Does there exist, for each d, a threshold h_d such that fullerenes with pentagon separation at least d exist for any number of hexagons ≥ h_d?
• RQ3 How many fullerenes with pentagon separation at least d exist for 1 ≤ d ≤ 5, up to 400 vertices?
• RQ4 Are the minimal fullerenes for a given d unique up to mirror image?
• RQ5 Can the structure of minimal fullerenes with high pentagon separation be characterized using Coxeter coordinates and patch-based constructions?

Key findings

• For odd d ≥ 3, the smallest fullerene with pentagon separation at least d is the icosahedral fullerene with Coxeter coordinates (⌈d/2⌉, ⌊d/2⌋) or (⌊d/2⌋, ⌈d/2⌋), with 15d² + 5 vertices.
• For even d, the unique smallest fullerene with pentagon separation at least d is the icosahedral fullerene with Coxeter coordinates (d/2, d/2), with 15d² vertices.
• For each d, there exists an h_d such that fullerenes with pentagon separation at least d exist for any number of hexagons ≥ h_d, generalizing prior results for h_1 and h_2.
• The number of fullerenes with pentagon separation at least d is computed up to 400 vertices for d = 1 to 5, with exact counts provided in Tables 3 and 4.
• For d = 5, no fullerenes with pentagon separation at least 5 exist with fewer than 380 vertices, and the first such fullerene appears at 380 vertices with 1,011,152,383 such fullerenes.
• The count of fullerenes with pentagon separation at least 5 reaches 1,784,313 at 400 vertices, indicating a rapid growth in the number of such stable configurations.