| Title: | The ▫$\sigma$▫-irregularity of trees with maximum degree ▫$5$▫ |
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| Authors: | ID Dimitrov, Darko (Author)
ID Kovijanić-Vukićević, Žana (Author)
ID Popivoda, Goran (Author)
ID Sedlar, Jelena (Author)
ID Škrekovski, Riste (Author)
ID Vujošević, Saša (Author) |
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| Files: |  URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0166218X25006857
 PDF - Presentation file, download (679,47 KB)
MD5: 376E48E44FF580DB7A28FE572333A1C8
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  RUDOLFOVO - Rudolfovo - Science and Technology Centre Novo Mesto
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| Abstract: | The ▫$\sigma$▫-irregularity, a variant of the well-established Albertson irregularity, is a topological invariant defined for a graph ▫$G=(V,E)$▫ as ▫$\sigma (G) = \sum_{u,v \in E}(d(u) - d(v))^2$▫, where ▫$d(u)$▫ and ▫$d(v)$▫ denote the degrees of vertices ▫$u$▫ and ▫$v$▫, respectively. Recent research has successfully characterized chemical trees with the maximum ▫$\sigma$▫-irregularity. In this paper, we expand upon this research by establishing several structural properties of maximal trees with prescribed maximum degree ▫$\Delta$▫. Application of these properties enables us to characterize maximal trees with ▫$\Delta = 5$▫. We establish that extremal trees contain only vertices of degrees ▫$1$▫, ▫$2$▫ and ▫$\Delta$▫. Moreover, the number of edges with both end-vertices having the degree ▫$2$▫ or ▫$\Delta$▫ is very small, so almost all edges have the (second) maximum possible contribution to ▫$\sigma$▫-irregularity. We believe this property or similar should extend to maximal trees for any value of ▫$\Delta$▫, so this is an interesting direction for further research. |
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| Keywords: | regular graph, trees, maximum degree, [sigma]-irregularity, maximal graphs, graph measure, topological index |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.03.2026 |
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| Publisher: | Elsevier |
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| Year of publishing: | 2026 |
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| Number of pages: | str. 124-136 |
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| Numbering: | Vol. 382 |
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| PID: | 20.500.12556/DiRROS-27405  |
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| UDC: | 519.17 |
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| ISSN on article: | 0166-218X |
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| DOI: | 10.1016/j.dam.2025.11.045 |
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| COBISS.SI-ID: | 266933507 |
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| Copyright: | © 2025 The Authors |
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| Note: | Žana Kovijanić Vukićević, Goran Popivoda, Jelena Sedlar, Riste Škrekovski, Saša Vujošević;
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| Publication date in DiRROS: | 05.02.2026 |
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| Views: | 57 |
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| Downloads: | 21 |
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