20.500.12556/RUP-23481 Some results on ▫$\sigma_t$▫-irregularity 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. σ_t-iregularnost (oziroma sigma total indeks) je grafovska invarianta, definirana kot [ \sigma_t(G)=\sum_{{u,v}\subseteq V(G)}(d(u)-d(v))^2, ] kjer (d(z)) označuje stopnjo vozlišča (z). To mero iregularnosti je leta 2019 predlagal Réti, leta 2023 pa sta jo ponovno uvedla Dimitrov in Stevanović. V članku opazimo, da velja (\sigma_t(G)=n^2\operatorname{Var}(G)), kjer je (\operatorname{Var}(G)) varianca stopenj grafa. Pokažemo, da ima med vsemi polnimi dvodelnimi grafi na (n) vozliščih največji indeks sigma total eden od ustreznih polnih dvodelnih grafov, katerih velikosti delov sta najbližje vrednostma (n(2-\sqrt{2})/4) in (n(2+\sqrt{2})/4). Poleg tega podamo različne zgornje in spodnje meje za σ_t-iregularnost. V tej smeri vzpostavimo tudi povezavo med energijo grafa (\mathcal{E}(G)) in σ_t-iregularnostjo ter izpeljemo meje, povezane z Laplaceovimi lastnimi vrednostmi grafa. irregularity total irregularity energy of graphs Laplacian eigenvalues nepravilnost iregularnost popolna nepravilnost energija grafov Laplaceove lastne vrednosti true false true Angleški jezik Slovenski jezik Članek v reviji 2026-08-14 11:04:27 2026-08-14 11:04:27 2026-08-15 03:06:55 0000-00-00 00:00:00 2026 0 0 str. 1-12 no. 4, [article no.] P4.01 Vol. 26 2026 0000-00-00 Zaloznikova NiDoloceno NiDoloceno 0000-00-00 0000-00-00 2026-08-11 519.17 1855-3974 10.26493/1855-3974.3268.60f 254097667 RAZ_Filipovski_Slobodan_2026.pdf RAZ_Filipovski_Slobodan_2026.pdf 1 1A83748BA5C7D5BD8A6561F989D45AB7 b07b5b890c64cbd6f586755a64a84234330cc704004ec6a910c364c700806815 0f9da738-97bf-11f1-9e8d-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=34443 https://amc-journal.eu/index.php/amc/article/view/3268 1 ed46d72c-97be-11f1-9e8d-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=34441 Fakulteta za matematiko, naravoslovje in informacijske tehnologije 0 0 0