Maximum cardinality resonant sets and maximal alternating sets of hexagonal systems

Klavar, Sandi; Salem, Khaled; Taranenko, Andrej

doi:10.1016/j.camwa.2009.06.011

Cited by 4 publications

(3 citation statements)

References 27 publications

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“…There is a high correlation between the Hosoya index and the boiling points of the acyclic alkanes. There is huge volume of literature on matchings of hexagonal systems [ 22 , 23 , 24 , 25 ]. Our objective of the paper is to identify similar properties of three nanotubes.…”

Section: Perfect Matching and Matching Ratiomentioning

confidence: 99%

Computational Aspects of Carbon and Boron Nanotubes

Manuel

2010

Molecules

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Carbon hexagonal nanotubes, boron triangular nanotubes and boron α-nanotubes are a few popular nano structures. Computational researchers look at these structures as graphs where each atom is a node and an atomic bond is an edge. While researchers are discussing the differences among the three nanotubes, we identify the topological and structural similarities among them. We show that the three nanotubes have the same maximum independent set and their matching ratios are independent of the number of columns. In addition, we illustrate that they also have similar underlying broadcasting spanning tree and identical communication behavior.

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Section: Perfect Matching and Matching Ratiomentioning

confidence: 99%

Computational Aspects of Carbon and Boron Nanotubes

Manuel

2010

Molecules

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“…By Clar's theory, the most important Kekulé structures are those in which the number of aromatic sextets equals the Clar number. Some well-investigated concepts that are closely related to Clar covers are resonant sets, alternating sets, and the Fries number (for some research on these topics see [6][7][8][9]).…”

Section: Introductionmentioning

confidence: 99%

The Multivariable Zhang–Zhang Polynomial of Phenylenes

Tratnik

2023

Axioms

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The Zhang–Zhang polynomial of a benzenoid system is a well-known counting polynomial that was introduced in 1996. It was designed to enumerate Clar covers, which are spanning subgraphs with only hexagons and edges as connected components. In 2018, the generalized Zhang–Zhang polynomial of two variables was defined such that it also takes into account 10-cycles of a benzenoid system. The aim of this paper is to introduce and study a new variation of the Zhang–Zhang polynomial for phenylenes, which are important molecular graphs composed of 6-membered and 4-membered rings. In our case, Clar covers can contain 4-cycles, 6-cycles, 8-cycles, and edges. Since this new polynomial has three variables, we call it the multivariable Zhang–Zhang (MZZ) polynomial. In the main part of the paper, some recursive formulas for calculating the MZZ polynomial from subgraphs of a given phenylene are developed and an algorithm for phenylene chains is deduced. Interestingly, computing the MZZ polynomial of a phenylene chain requires some techniques that are different to those used to calculate the (generalized) Zhang–Zhang polynomial of benzenoid chains. Finally, we prove a result that enables us to find the MZZ polynomial of a phenylene with branched hexagons.

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“…A fullerene is a molecule entirely composed of carbon. Fullerenes have wide application in various fields including electronic and optic engineering [2,3] medical science [4] and biotechnology [5] and have received a lot of recent chemists and mathematicians' attention [6,7]. From a mathematical point of view, every skeletal structure of a chemical molecule can be represented by a graph called molecular graph in which vertices indicate atoms and edges represent chemical bonds between atoms.…”

Section: Introductionmentioning

confidence: 99%

A mathematical programming model for computing the Fries number of a fullerene

Salami

Ahmadi

2015

Applied Mathematical Modelling

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a b s t r a c tA fullerene graph is a cubic 3-connected plane graph with pentagonal and hexagonal faces. The Fries number of a fullerene is the maximum number of benzene-like faces over all possible perfect matchings. The Fries number and its associated Kekulé structure of a fullerene play a key role in molecular energy and stability. In this paper we propose a binary integer linear programming and a quadratic programming model for determining the Fries number of a fullerene. Moreover, interior point approach, as one of the most robust optimization techniques, is implemented to find the optimal solution of the proposed quadratic programming problem in moderate computing time.

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Maximum cardinality resonant sets and maximal alternating sets of hexagonal systems

Cited by 4 publications

References 27 publications

Computational Aspects of Carbon and Boron Nanotubes

Computational Aspects of Carbon and Boron Nanotubes

The Multivariable Zhang–Zhang Polynomial of Phenylenes

A mathematical programming model for computing the Fries number of a fullerene

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