A Curious Family of Convex Benzenoids and Their Altans

IF 1 Q1 MATHEMATICS
N. Bašić, P. Fowler
{"title":"A Curious Family of Convex Benzenoids and Their Altans","authors":"N. Bašić, P. Fowler","doi":"10.47443/dml.2021.s218","DOIUrl":null,"url":null,"abstract":"The altan graph of G , a ( G, H ) , is constructed from graph G by choosing an attachment set H from the vertices of G and attaching vertices of H to alternate vertices of a new perimeter cycle of length 2 | H | . When G is a polycyclic plane graph with maximum degree 3 , the natural choice for the attachment set is to take all perimeter degree- 2 vertices in the order encountered in a walk around the perimeter. The construction has implications for the electronic structure and chemistry of carbon nanostructures with molecular graph a ( G, H ) , as kernel eigenvectors of the altan correspond to non-bonding π molecular orbitals of the corresponding unsaturated hydrocarbon. Benzenoids form an important subclass of carbon nanostructures. A convex benzenoid has a boundary on which all vertices of degree 3 have exactly two neighbours of degree 2 . The nullity of a graph is the dimension of the kernel of its adjacency matrix. The possible values for the excess nullity of a ( G, H ) over that of G are 2 , 1 , or 0 . Moreover, altans of benzenoids have nullity at least 1 . Examples of benzenoids where the excess nullity is 2 were found recently. It has been conjectured that the excess nullity when G is a convex benzenoid is at most 1 . Here, we exhibit an infinite family of convex benzenoids with 3 -fold dihedral symmetry (point group D 3h ) where nullity increases from 2 to 3 under altanisation. This family accounts for all known examples with the excess nullity of 1 where the parent graph is a singular convex benzenoid.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics Letters","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.47443/dml.2021.s218","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1

Abstract

The altan graph of G , a ( G, H ) , is constructed from graph G by choosing an attachment set H from the vertices of G and attaching vertices of H to alternate vertices of a new perimeter cycle of length 2 | H | . When G is a polycyclic plane graph with maximum degree 3 , the natural choice for the attachment set is to take all perimeter degree- 2 vertices in the order encountered in a walk around the perimeter. The construction has implications for the electronic structure and chemistry of carbon nanostructures with molecular graph a ( G, H ) , as kernel eigenvectors of the altan correspond to non-bonding π molecular orbitals of the corresponding unsaturated hydrocarbon. Benzenoids form an important subclass of carbon nanostructures. A convex benzenoid has a boundary on which all vertices of degree 3 have exactly two neighbours of degree 2 . The nullity of a graph is the dimension of the kernel of its adjacency matrix. The possible values for the excess nullity of a ( G, H ) over that of G are 2 , 1 , or 0 . Moreover, altans of benzenoids have nullity at least 1 . Examples of benzenoids where the excess nullity is 2 were found recently. It has been conjectured that the excess nullity when G is a convex benzenoid is at most 1 . Here, we exhibit an infinite family of convex benzenoids with 3 -fold dihedral symmetry (point group D 3h ) where nullity increases from 2 to 3 under altanisation. This family accounts for all known examples with the excess nullity of 1 where the parent graph is a singular convex benzenoid.
一个奇特的凸类苯族及其Altans
图G的altgraph a (G, H)是由图G通过从G的顶点中选择一个附属集H,并将H的顶点附加到一个新的周长为2b| H |的交替顶点上来构造的。当G是一个最大度数为3的多环平面图时,附件集的自然选择是取所有周长度数为- 2的顶点,按照在周长周围行走时遇到的顺序。该结构对具有分子图a (G, H)的碳纳米结构的电子结构和化学性质具有启示意义,因为它的核特征向量对应于相应不饱和烃的非键π分子轨道。苯类化合物是碳纳米结构的一个重要亚类。凸苯类有一个边界,在这个边界上,所有3次顶点都恰好有两个2次顶点相邻。图的零是它的邻接矩阵核的维数。a (G, H)超过G的多余零值的可能值是2、1或0。而且,苯类化合物的偶联至少为1。最近发现了苯类化合物的过量零值为2的例子。我们推测,当G是凸苯类时,其多余零点不超过1。在这里,我们展示了一个具有3重二面体对称(点群d3h)的凸苯类无穷族,在交替作用下,零从2增加到3。这个族解释了所有已知的超零为1的例子,其中父图是一个奇异凸苯类。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Discrete Mathematics Letters
Discrete Mathematics Letters Mathematics-Discrete Mathematics and Combinatorics
CiteScore
1.50
自引率
12.50%
发文量
47
审稿时长
12 weeks
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信