20.500.12556/RUL-185661 Some results on $\sigma_t$-irregularity The $\sigma_t$-irregularity (or sigma total index) is a graph invariant which is defined as $\sigma_t(G) = \sum_{\{u,v\}\subseteq V(G)}(d(u)-d(v))^2$, where $d(z)$ denotes the degree of $z$. 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, either the complete bipartite graph $K_{\lfloor{n \over 4}(2-\sqrt{2})\rfloor, \lceil {n \over 4}(2+\sqrt{2})\rceil}$ or $K_{\lceil{n \over 4}(2-\sqrt{2})\rceil, \lfloor{n \over 4}(2+\sqrt{2})\rfloor}$ has the maximum sigma total index. Moreover, various upper and lower bounds for $\sigma_t$-irregularity are provided; in this direction we give a relation between the graph energy ${\mathcal E}(G)$ and sigma total index $\sigma_t(G)$ and give new proofs of two results by Dimitrov and Stevanović. Applying Fiedler’s characterization of the largest and the second smallest Laplacian eigenvalue of the graph, we also establish new relationships between $\sigma_t$ and $\sigma$. We conclude the paper with two conjectures. 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 2026-08-17 09:00:43 2026-08-17 09:00:49 2026-08-18 04:04:44 0000-00-00 00:00:00 2026 0 0 Str. 1-12 no. 4, art. P4.01 Vol. 26 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.17 1855-3974 10.26493/1855-3974.3268.60f 254097667 https://amc-journal.eu/index.php/amc/article/view/3268 1 01f0f36b-9a09-11f1-9b0d-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=244408 amc_3268_(2).pdf amc_3268_(2).pdf 1 1A83748BA5C7D5BD8A6561F989D45AB7 b07b5b890c64cbd6f586755a64a84234330cc704004ec6a910c364c700806815 d8d22522-9a0c-11f1-9b0d-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=244410 Fakulteta za matematiko in fiziko 0 0 0