Уопштења Шатенових норми графова и комбинаторне примене

Abstract

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

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 combinatorial theory 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. Energy of 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 the special cases of this norm for p = 1 corresponds to the Energy of graph. In chapter three of this dissertation we are given the most original scientic contribution. We prove the conjecture of Nikiforov about Schatten norms of graphs when p > 2. First we prove that conjecture is true for 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 the proof of this conjecture is also an original result which gives a sharp upper bound of sum of quadratic of the largest k singular values of graph. A corollary of this theorem which gives an upper bound for sum of squares of the biggest two singular values of graph can be useful in further research. In the subsection 3.3 we give an original theorem about asymptotic properties of spectrum and thus energy of complement graph for a large values of n. In the subsection 3.4 we calculate a mean of p-th degree of singular values and upper bound of geometric mean of almost all graphs. The last chapter shows relation between combinatorial theory of graphs with universal universal algebra and mathematical logic. The central part of this chapter is original and shorter proof of an important theorem which solves a dichotomy conjecture for CSP problem on undirected graphs.