Ekviangularni pravci

master's thesis

• Title (english): Equiangular lines
• Author: Ema Požek
• Mentor: Vedran Krčadinac (mentor)
• Committee member: Vedran Krčadinac (predsjednik povjerenstva)
• Committee member: Igor Ciganović (član povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-11-30, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu obrađeni su ekviangularni pravci. Interpretirani su s gledišta algebarske teorije grafova. Dokazane su apsolutna i relativna ocjena za veličinu ekviangularnog skupa. Definirani su operacija prespajanja, klase prespajanja, grafovi prespajanja te dvografi. Primjerima i strogo matematički pokazane su bijekcije između klasa prespajanja, grafova prespajanja i dvografa. Uveden je pojam regularnog dvografa i opisana njegova važnost u proučavanju ekviangularnih pravaca. Dokazana su osnovna svojstva jako regularnih grafova. S pomoću primjera opisani su Paleyevi grafovi kao klasa jako regularnih grafova. Opisana je veza izmedu jako regularnih grafova i regularnih dvografa. Navedeni su mogući parametri jako regularnih grafova i regularnih grafova prespajanja. Opisana je metoda pronalaska optimalnog ekviangularnog skupa s pomoću Paleyevih grafova. Koristeći Paley(17) konstruiran je ekviangularan skup u \(\R^9\).

Abstract (english)

In this thesis equiangular lines are discussed. They are interpreted in terms of algebraic graph theory. The absolute bound and the relative bound for maximum number of equiangular lines are proved. Switching, switching classes, switching graphs and two-graphs are defined. Rigorously, as well as through examples, the one-to-one correspondence between switching classes, switching graphs and two-graphs is demonstrated. Regular two-graphs and their significance in the analysis of equiangular lines are introduced. The basic properties of strongly regular graphs are proved. Through examples, Paley graphs are described as a class of strongly regular graphs. The relation between strongly regular graphs and regular two-graphs is depicted. Feasible parameters of strongly regular graphs and regular switching graphs are listed. The method of finding the optimal equiangular set using Paley graphs is described. Using Paley(17), an equiangular set in \(\R^9\) is constructed.

• Keywords: algebarska teorija grafova; operacija prespajanja; klase prespajanja; grafovi prespajanja; dvografi; regularni dvograf; jako regularnih grafovi; Paleyevi grafovi
• Keywords (english): algebraic graph theory; switching; switching classes; switching graphs; two-graphs; regular two-graphs; strongly regular graphs; Paley graphs
• Language: croatian
• URN:NBN: urn:nbn:hr:217:457297
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access; Embargo expiration date: 2023-12-19
• Created on: 2022-12-19 12:37:32