Naslov (srp)

Уопштења Шатенових норми графова и комбинаторне примене : докторска дисертација

Autor

Lazarević, Ivan, 1988-

Doprinosi

Mateljević, Miodrag, 1960-
Božin, Vladimir, 1949-
Erić, Aleksandra
Stanić, Zoran, 1967-
Jocić, Danko, 1975-

Opis (srp)

У овој докторској дисертацији добијени су неки резултети у области теорије графова и њених примена.

Opis (srp)

Математика - Примењена математика / mathematics - аpplied mathematics Datum odbrane: 12.07.2022.

Opis (eng)

In this doctoral thesis we obtained some results in graph theory and its applica-tions. In the rst chapter, we give the review of basic notions and theorems of combinatorialtheory of graphs, spectral theory of graphs, random graphs and distribution of their eigenvalues.The most detailed consideration is given to adjacency matrix and properties of its spectrum.In particular, in this dissertation we study Energy of graphs and generalizations of it. Energyof graph is the sum of absolute values of eigenvalues of a graph.Schatten norms of graphs represent p-th degree norm of singular values of graph, and thespecial cases of this norm for p = 1 corresponds to the Energy of graph. In chapter three of thisdissertation we are given the most original scientic contribution. We prove the conjecture ofNikiforov about Schatten norms of graphs when p > 2. First we prove that conjecture is truefor some special classes of graph (for trees and strongly regular graph with maximal energy).After that, we prove the conjecture in the general case. Auxiliary theorem obtained in theproof of this conjecture is also an original result which gives a sharp upper bound of sum ofquadratic of the largest k singular values of graph. A corollary of this theorem which gives anupper bound for sum of squares of the biggest two singular values of graph can be useful infurther research. In the subsection 3.3 we give an original theorem about asymptotic propertiesof spectrum and thus energy of complement graph for a large values of n. In the subsection3.4 we calculate a mean of p-th degree of singular values and upper bound of geometric meanof almost all graphs.The last chapter shows relation between combinatorial theory of graphs with universaluniversal algebra and mathematical logic. The central part of this chapter is original and shorterproof of an important theorem which solves a dichotomy conjecture for CSP problem on undirectedgraphs.

Jezik

srpski

Datum

2022

Licenca

Ovo delo je licencirano pod uslovima licence
Creative Commons CC BY-NC-ND 3.0 AT - Creative Commons Autorstvo - Nekomercijalno - Bez prerada 3.0 Austria License.

http://creativecommons.org/licenses/by-nc-nd/3.0/at/legalcode

Predmet

OSNO - Opšta sistematizacija naučnih oblasti, Kombinatorna analiza. Teorija grafova

Енергија графа, сопствене вредности, сингуларне вредности, Шатенове норме, случајни граф, CSP проблем

OSNO - Opšta sistematizacija naučnih oblasti, Kombinatorna analiza. Teorija grafova

Energy of graph, eigenvalues, singular values, Schatten norms, random graph, CSP problem.