The Number of Perfect Matchings in (3,6)-Fullerene

Pub Date : 2023-06-01 DOI:10.1051/wujns/2023283192
Rui Yang, Mingzhu Yuan
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引用次数: 0

Abstract

A [see formula in PDF]-fullerene is a connected cubic plane graph whose faces are only triangles and hexagons, and has the connectivity [see formula in PDF] or [see formula in PDF]. The [see formula in PDF]-fullerenes with connectivity [see formula in PDF] are the tubes consisting of [see formula in PDF] concentric hexagonal layers such that each layer consists of two hexagons, capped on each end by two adjacent triangles, denoted by [see formula in PDF]. A [see formula in PDF]-fullerene [see formula in PDF] with [see formula in PDF] vertices has exactly [see formula in PDF] perfect matchings. The structure of a [see formula in PDF]-fullerene [see formula in PDF] with connectivity [see formula in PDF] can be determined by only three parameters [see formula in PDF], [see formula in PDF] and[see formula in PDF], thus we denote it by [see formula in PDF], where [see formula in PDF] is the radius (number of rings), [see formula in PDF] is the size (number of spokes in each layer, [see formula in PDF], [see formula in PDF] is even), and [see formula in PDF] is the torsion ([see formula in PDF]). In this paper, the counting formula of the perfect matchings in [see formula in PDF]is given, and the number of perfect matchings is obtained. Therefore, the correctness of the conclusion that every bridgeless cubic graph with [see formula in PDF] vertices has at least [see formula in PDF] perfect matchings proposed by Esperet et al is verified for [see formula in PDF]-fullerene [see formula in PDF].
(3,6)-富勒烯的完全匹配数
富勒烯是一个连通的三次平面图形,它的面只有三角形和六边形,并且具有连通性[见PDF中的公式]或[见PDF中的公式]。具有连通性的富勒烯是由[见PDF中的公式]同心圆六边形层组成的管,这样每一层由两个六边形组成,每一端由两个相邻的三角形覆盖,用[见PDF中的公式]表示。一个[见PDF公式]-富勒烯[见PDF公式]与[见PDF公式]顶点完全匹配[见PDF公式]。(见公式的结构以PDF]富勒烯(见公式以PDF)与连接(见公式以PDF)只能由三个参数(见公式以PDF),(见公式以PDF)和(见公式以PDF),因此我们表示(见公式以PDF),其中(见公式以PDF)是半径(环),(见公式以PDF)是大小(每一层的辐条数量,(见公式以PDF),(见公式以PDF)甚至),和[见PDF公式]是扭转(见PDF公式])。本文给出了[公式见PDF]中完美匹配的计数公式,并得到了完美匹配的个数。因此,对于[见公式]-富勒烯[见公式],验证了Esperet等人提出的每个顶点为[见公式]的无桥三次图至少具有[见公式]个完美匹配的结论的正确性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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