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Creator:	Ivan Vanja Boroja
Title:	Compression of the commuting graph of rings and other algebraic structures
Copyright date:	2025
Permanent link:	https://phaidra.unibl.org/detail_object/o:3511
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Select license:	Autorstvo-Nekomercijalno-Deliti pod istim uslovima 3.0 (CC BY-NC-SA 3.0)
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Academic metadata

Theses Type:	Doktorska disertacija
Academic Expertise :	Prirodno-matematičke nauke
Academic Title:	-
University:	Univerzitet u Banjoj Luci
Faculty:	Prirodno-matematički fakultet

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Alternative title:	Compression of the commuting graph of rings and other algebraic structures
description:	Prirodne nauke Matematika
description:	Datum odbrane:18.07.2025.
Other responsibilities:	Bojan Nikolić (predsednik komisije)
Other responsibilities:	Vladimir Božović (član komisije)
Other responsibilities:	Damjana Kokol Bukovšek (član komisije)
Other responsibilities:	Dimitrije Čvokić (član komisije)
Abstract:	In thisthesisweinvestigatetherecentlyintroducedcompressedcommutinggraph Λ1(R) of aunitalring R. Thisisagraphwhoseverticesareequivalenceclassesofelementsof R according totherelation ∼ whichisdefinedas a ∼ b if andonlyif a and b generate the same unitalsubring.Twoverticesareconnectedbyanedgeifandonlyiftheirrepresentatives commute.Thisgraphcanbeseenasacompressionoftheregularcommutinggraph Γ(R). We proveinthethesisthatformatrixalgebrasoverfinitefieldsthiscompressionisthebestpossible compression thatinducesafunctorfromthecategoryofunitalringstothecategoryofgraphs. Wealsodiscusssomepropertiesofthisgraph,forexample,thegraphgivesinformationabout the setofunitalsubringsof R generated byoneelement.Thisviewwasappliedinourresult characterizinginfiniteunitalringswithonlyfinitelymanyunitalsubrings. In ourrecentarticlewewereabletocompletelydescribethegraph Λ1(M2(F)) for afinite field F. Themaincontributionofthisthesisisthecompletedescriptionofthegraph Λ1(M3(F)) for aprimefield F = GF(p). Toachievethisgoalwecombinedmethodsfromfieldtheory, projectivegeometryandcombinatorics.Wefirstdescribethesetofvertices,relyingonthe Jordan formofmatrices,andthendeterminethestructureoftheneighborhoodofeachvertex. The corepartofthegraphisthendescribedusingabijectivecorrespondencewithapoint-line pairs intheprojectiveplaneover GF(p). Inaddition,wealsogiveashortalgorithmthatcanbe used toconstruct Λ1(M3(GF(p))). As a consequence of our result we are also able to describe the graph Γ(M3(GF(p)) using theso-called"blow-up"process.The description of this graph was an open problem for several years. wasanopenproblemforseveralyears.
Authors Key words:	Commuting graph,compressedcommuting graph,matrix ring,finite field
Language:	English

Abstract:

In thisthesisweinvestigatetherecentlyintroducedcompressedcommutinggraph Λ1(R) of aunitalring R. Thisisagraphwhoseverticesareequivalenceclassesofelementsof R according totherelation ∼ whichisdefinedas a ∼ b if andonlyif a and b generate the same unitalsubring.Twoverticesareconnectedbyanedgeifandonlyiftheirrepresentatives commute.Thisgraphcanbeseenasacompressionoftheregularcommutinggraph Γ(R). We proveinthethesisthatformatrixalgebrasoverfinitefieldsthiscompressionisthebestpossible compression thatinducesafunctorfromthecategoryofunitalringstothecategoryofgraphs. Wealsodiscusssomepropertiesofthisgraph,forexample,thegraphgivesinformationabout the setofunitalsubringsof R generated byoneelement.Thisviewwasappliedinourresult characterizinginfiniteunitalringswithonlyfinitelymanyunitalsubrings. In ourrecentarticlewewereabletocompletelydescribethegraph Λ1(M2(F)) for afinite field F. Themaincontributionofthisthesisisthecompletedescriptionofthegraph Λ1(M3(F)) for aprimefield F = GF(p). Toachievethisgoalwecombinedmethodsfromfieldtheory, projectivegeometryandcombinatorics.Wefirstdescribethesetofvertices,relyingonthe Jordan formofmatrices,andthendeterminethestructureoftheneighborhoodofeachvertex. The corepartofthegraphisthendescribedusingabijectivecorrespondencewithapoint-line pairs intheprojectiveplaneover GF(p). Inaddition,wealsogiveashortalgorithmthatcanbe used toconstruct Λ1(M3(GF(p))). As a consequence of our result we are also able to describe the graph Γ(M3(GF(p)) using theso-called"blow-up"process.The description of this graph was an open problem for several years. wasanopenproblemforseveralyears.