New methods of calculating chromatic polynomials

The problem of coloring graphs discussed in this thesis constitutes one of the important sub-branches of graph theory, one of the most growing areas in recent years. In this study, the theory of coloring, which can be expressed as the least how many colors the vertices of a graph can be painted, provided that two adjacent vertices are painted with different colors, in the graph theory that arose as a result of the search for the answer to a question posed in 1736, is discussed. Coloring graphs is an important field in graph theory, which is the fastest growing branch of mathematics in recent years. Coloring the corners can be done in different ways. Similarly, it is possible to color edges and faces. All these colorings have different applications. In a sense, the coloring of the vertices is analogous to the problem of labeling a graph. The number of all colorings of a graph is called the chromatic number of graph. The resulting polynomial in the coloring problem will be called the chromatic polynomial of a graph. In this thesis, chromatic polynomials of various graphs are discussed. The first chapter of this thesis consistsing of five chapters is the introductory part. In this section, the concepts that will be used later are introduced. In addition, the previously proven chromatic polynomial calculation methods are mentioned. In the second chapter, the theoretical foundations are given. In chapter 3, the materials and methods used in the thesis are mentioned. In the fourth chapter, by separating a given connected graph into smaller graphs with certain methods, new methods are obtained for calculating the chromatic polynomials of these graphs. The chromatic polynomials of graphs combined in different ways have been obtained by means of the Birkhoff-Lewis theorem, corner and edge separation. In the fifth and last part, the findings of the thesis were discussed and a general evaluation was made.