Počet neizomorfních grafů s daným počtem vrcholů a hran

Abstract

In this text we find out how many \textit{nonisomorphic} graphs are there on $n$ vertices. Using relatively simple combinatorial reasoning it is possible to determine how many \textit{different} graphs are there on given set of vertices. But many of them have the same structure. We say about two graphs with the same structure that they are \textit{isomorphic}. If two graphs are isomorphic, then we can just relabel vertices of one of them to get two identical graphs. From the above it follows that more interesting question than how many different graphs are there on $n$ vertices is the question how many graphs with different structure, thus nonisomorphic graphs, are there on $n$ vertices. Here, however, simple combinatorial reasoning is no longer sufficient. We have to use some results of Pólya's enumeration theory. The character of this thesis is compilatory, thus the thesis summarizes what is known about this problem so the text can be understood as an educational text. There are also lots of examples, algorithms, theorems and proofs that were elaborated completely individually.