| Title: | Some results on ▫$\sigma_t$▫-irregularity |
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| Authors: | ID Filipovski, Slobodan (Author)
ID Dimitrov, Darko (Author)
ID Knor, Martin (Author)
ID Škrekovski, Riste (Author) |
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| Files: |  RAZ_Filipovski_Slobodan_2026.pdf (335,30 KB)
MD5: 1A83748BA5C7D5BD8A6561F989D45AB7
 https://amc-journal.eu/index.php/amc/article/view/3268
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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: | FAMNIT - Faculty of Mathematics, Science and Information Technologies
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| Abstract: | The (\sigma_t)-irregularity (or sigma total index) is a graph invariant defined as [ \sigma_t(G)=\sum_{{u,v}\subseteq V(G)}(d(u)-d(v))^2, ] where (d(z)) denotes the degree of a vertex (z). This irregularity measure was proposed by Réti in 2019 and recently rediscovered by Dimitrov and Stevanović in 2023. In this paper, we remark that (\sigma_t(G)=n^2\operatorname{Var}(G)), where (\operatorname{Var}(G)) is the degree variance of the graph. We show that among all complete bipartite graphs on (n) vertices, one of the corresponding complete bipartite graphs whose part sizes are closest to (n(2-\sqrt{2})/4) and (n(2+\sqrt{2})/4) has the maximum sigma total index. Moreover, various upper and lower bounds for (\sigma_t)-irregularity are provided. In this direction, we establish a relation between the graph energy (\mathcal{E}(G)) and (\sigma_t)-irregularity and derive bounds related to the Laplacian eigenvalues of the graph. |
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| Keywords: | irregularity, total irregularity, energy of graphs, Laplacian eigenvalues |
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| Publication version: | Version of Record |
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| Publication date: | 11.08.2026 |
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| Year of publishing: | 2026 |
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| Number of pages: | str. 1-12 |
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| Numbering: | Vol. 26, no. 4, [article no.] P4.01 |
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| PID: | 20.500.12556/RUP-23481  |
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| UDC: | 519.17 |
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| ISSN on article: | 1855-3974 |
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| DOI: | 10.26493/1855-3974.3268.60f |
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| COBISS.SI-ID: | 254097667 |
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| Publication date in RUP: | 14.08.2026 |
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| Views: | 30 |
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| Downloads: | 2 |
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