Coronoids, patches and generalised altans

In 2012, a family of benzenoids was introduced by Cruz, Gutman, and Rada, which they called convex benzenoids. In this paper we introduce the convexity deficit, a new topological index intended for benzenoids and, more generally, fusenes. This index measures by how much a given fusene departs from convexity. It is defined in terms of the boundary-edges code. In particular, convex benzenoids are exactly the benzenoids having convexity deficit equal to 0. Quasi-convex benzenoids form the family of non-convex benzenoids that are closest to convex, i.e., they have convexity deficit equal to 1. Finally, we investigate convexity deficit of several important families of benzenoids.

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“…[11] This family was further studied and enumerated in. [5] A convex benzenoid can be characterised via its boundary-edges code.…”

Section: Convex Benzenoidsmentioning

confidence: 99%

Convexity Deficit of Benzenoids

Bašić

Berkemer

Fallmann

et al. 2020

Croat. chem. acta (Online)

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“…A comprehensive bibliography on the theory of coronoids can be found in books [24,34]. A mathematical formalization of the theory of coronoids and related structures was developed in [35]. The numbers of helicenic hexagonal chains were reported in [36,37].…”

Section: Classes Of Annulimentioning

confidence: 99%

Enumeration of Pentahexagonal Annuli in the Plane

Dobrynin

Rosenfeld

2019

Mathematics

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Pentahexagonal annuli are closed chains consisting of regular pentagons and hexagons. Such configurations can be easily recognized in various complex designs, in particular, in molecular carbon constructions. Results of computer enumeration of annuli without overlapping on the plane are presented for up to 18 pentagons and hexagons. We determine how many annuli have certain properties for a fixed number of pentagons. In particular, we consider symmetry, pentagon separation (the least ring-distance between pentagons), uniformity of pentagon distribution, and pentagonal thickness (the size of maximal connected part of pentagons) of annuli. Pictures of all annuli with the number of pentagons and hexagons up to 17 are presented (more than 1300 diagrams).

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“…Hereinafter, the word benzenoid is used as a synonym for benzenoid graph. We build upon theoretical treatment of benzenoids from [2]. The framework of [2] allows for generalisation to structures called infinite benzenoids.…”

Section: Introductionmentioning

confidence: 99%

“…We build upon theoretical treatment of benzenoids from [2]. The framework of [2] allows for generalisation to structures called infinite benzenoids. In 2012 a family of convex benzenoids was introduced by Cruz, Gutman and Rada [4].…”

Section: Introductionmentioning

confidence: 99%

Infinite benzenoids

Bašić

2019

Art Discrete Appl. Math.

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A family of benzenoids, called convex benzenoids, was introduced in 2012 by Cruz, Gutman and Rada. In a later paper by the present author et al., several equivalent characterisations of convex benzenoids have been given and their equivalence was proved. Along the way an infinite benzenoid called the half-plane was used for the purpose of theoretical reasoning. In this short paper, some properties of infinite benzenoids are discussed. It is proved that their boundary consists of countably many connected components. Convex infinite benzenoids are classified and it is proved that there are only countably many convex infinite benzenoids, whilst there are uncountably many infinite (non-convex) benzenoids. We also show that there are countably many infinite benzenoids which have a finite number of 1s in their boundary-edges code.

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Coronoids, patches and generalised altans

Cited by 11 publications

References 27 publications

Convexity Deficit of Benzenoids

Convexity Deficit of Benzenoids

Enumeration of Pentahexagonal Annuli in the Plane

Infinite benzenoids

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