| Title: | The multivariable Zhang-Zhang polynomial of phenylenes |
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| Authors: | ID Tratnik, Niko (Author) |
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| Files: |  Tratnik-2023-The_Multivariable_Zhang-Zhang_Pol.pdf (719,26 KB)
MD5: 5934C26509350FB4EF432B001A85A7ED
 https://doi.org/10.3390/axioms12111053
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| Language: | English |
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| Work type: | Article |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | FNM - Faculty of Natural Sciences and Mathematics
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| Abstract: | 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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| Keywords: | Zhang-Zhang polynomial, phenylene, Clar cover, Kekulé structure |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Submitted for review: | 06.10.2023 |
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| Article acceptance date: | 14.11.2023 |
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| Publication date: | 15.11.2023 |
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| Publisher: | MDPI |
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| Year of publishing: | 2023 |
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| Number of pages: | Str. 1-18 |
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| Numbering: | Letn. 12, Št. 11, št. članka 1053 |
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| PID: | 20.500.12556/DKUM-87024  |
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| UDC: | 519.17 |
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| ISSN on article: | 2075-1680 |
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| COBISS.SI-ID: | 172554499 |
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| DOI: | 10.3390/axioms12111053 |
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| Publication date in DKUM: | 09.02.2024 |
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| Views: | 470 |
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| Downloads: | 50 |
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| Metadata: |    |
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| Categories: | Misc.
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