335 KB28 KB192 KB6 KB2026Dimitrov, DarkoFilipovski, SlobodanKnor, MartinŠkrekovski, Risteletnik:26številka:4, article p4.01str. 1-12DOI:10.26493/1855-3974.3268.60fISSN:1855-3974COBISSID_HOST:254097667URN:URN:NBN:SI:doc-IIYPYW2PenUniverza na PrimorskemArs mathematica contemporaneaenergija grafovenergy of graphsiregularnostirregularityLaplaceove lastne vrednostiLaplacian eigenvaluesnepravilnostpopolna nepravilnosttotal irregularitySome results on sigmasubt-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?$\sigma_t$?-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 ?${n \over 4}(2-\sqrt{2})$? in ?${n \over 4}(2+\sqrt{2})$?. Poleg tega podamo različne zgornje in spodnje meje za ?$\sigma_t$?-iregularnost. V tej smeri vzpostavimo tudi povezavo med energijo grafa ?${\mathcal E}(G)$? in ?$\sigma_t$?-iregularnostjo ter izpeljemo meje, povezane z Laplaceovimi lastnimi vrednostmi grafaTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije