# 11th meeting of the International Academy of Mathematical Chemistry

18-21 June 2015
Kranjska Gora, Slovenia
Europe/Ljubljana timezone
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# On the Clar number of fullerene graphs

Presented by Prof. Heping ZHANG
Type: Oral presentation

## Content

A {\em fullerene graph} is the molecular graph of a carbon fullerene molecule, whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds between pairs of atoms. In mathematics, a fullerene graph is a planar (or spherical) cubic graph with exactly 12 pentagonal faces and other hexagonal faces. The Clar number of a fullerene graph is the maximum number of mutually resonant disjoint hexagons in the fullerene. Zhang and Ye ever showed that that the Clar number of a fullerene graph with $n$ vertices is bounded above by $\lfloor n/6\rfloor-2$, where $\lfloor x\rfloor$ represents the largest integer not greater than $x$. We cannot find any fullerene graph with $n\equiv 2\pmod 6$ vertices attaining this bound. In fact we prove that the Clar number of such fullerene graphs is bounded above by $\lfloor n/6\rfloor-3$. Two experimentally produced fullerenes C$_{80}$:1(D$_{5d}$) and C$_{80}$:2(D$_{2}$) attain the bound $\lfloor n/6\rfloor-3$. Furthermore we prove that there is at least one fullerene graph attaining the improved bound in the case $n\equiv 2\pmod 6$ and the old bound for the other cases for every even number of vertices $n\geq20$ except $n=22$ and $n=30$.

## Place

Location: Kranjska Gora, Slovenia
Address: Ramada resort hotel Borovška cesta 99 4280 Kranjska Gora
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