|
Abstract:
|
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 scienti c 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. |