In this thesis we investigate different domination-related graph polynomials, like the connected domination polynomial, the independent domination polynomial, and the total domination polynomial. We prove some basic properties of these polynomials and obtain formulas for the calculation in special graph classes. Furthermore, we also prove results about the calculation of the different graph polynomials in product graphs and different representations of the graph polynomials. One focus of this thesis lays on the generalization of domination-related polynomials. In this context the trivariate domination polynomial is defined and some results about the bipartition polynomial, which is also a generalization of the domination polynomial, is presented. These two polynomials have many useful properties and interesting connections to other graph polynomials. Furthermore, some more general domination-related polynomials are defined in this thesis, which shows some possible directions for further research. / In dieser Dissertation werden verschiedene, zum Dominationspolynom verwandte, Graphenpolynome, wie das zusammenhängende Dominationspolynom, das unabhängige Dominationspolynom und das totale Dominationspolynom, untersucht. Es werden grundlegende Eigenschaften erforscht und Sätze für die Berechnung dieser Polynome in speziellen Graphenklassen bewiesen. Weiterhin werden Ergebnisse für die Berechnung in Produktgraphen und verschiedene Repräsentationen für diese Graphenpolynome gezeigt. Ein Fokus der Dissertation liegt auf der Verallgemeinerung der verschiedenen Dominationspolynome. In diesem Zusammenhang wird das trivariate Dominationspolynom definiert. Außerdem werden Ergebnisse für das Bipartitionspolynom bewiesen. Diese beiden Polynome haben viele interessante Eigenschaften und Beziehungen zu anderen Graphenpolynomen. Darüber hinaus werden weitere multivariate Graphenpolynome definiert, die eine mögliche Richtung für weitere Forschung auf diesem Gebiet aufzeigen.

Graph polynomials are polynomials associated to graphs that encode the number of subgraphs with given properties. We list different frameworks used to define graph polynomials in the literature. We present the edge elimination polynomial and introduce several graph polynomials equivalent to it. Thereby, we connect a recursive definition to the counting of colorings and to the counting of (spanning) subgraphs. Furthermore, we define a graph polynomial that not only generalizes the mentioned, but also many of the well-known graph polynomials, including the Potts model, the matching polynomial, the trivariate chromatic polynomial and the subgraph component polynomial. We proof a recurrence relation for this graph polynomial using edge and vertex operation. The definitions and statements are given in such a way that most of them are also valid in the case of hypergraphs.

Graph polynomials are polynomials associated to graphs that encode the number of subgraphs with given properties. We list different frameworks used to define graph polynomials in the literature. We present the edge elimination polynomial and introduce several graph polynomials equivalent to it. Thereby, we connect a recursive definition to the counting of colorings and to the counting of (spanning) subgraphs. Furthermore, we define a graph polynomial that not only generalizes the mentioned, but also many of the well-known graph polynomials, including the Potts model, the matching polynomial, the trivariate chromatic polynomial and the subgraph component polynomial. We proof a recurrence relation for this graph polynomial using edge and vertex operation. The definitions and statements are given in such a way that most of them are also valid in the case of hypergraphs.

In this thesis we investigate different domination-related graph polynomials, like the connected domination polynomial, the independent domination polynomial, and the total domination polynomial. We prove some basic properties of these polynomials and obtain formulas for the calculation in special graph classes. Furthermore, we also prove results about the calculation of the different graph polynomials in product graphs and different representations of the graph polynomials. One focus of this thesis lays on the generalization of domination-related polynomials. In this context the trivariate domination polynomial is defined and some results about the bipartition polynomial, which is also a generalization of the domination polynomial, is presented. These two polynomials have many useful properties and interesting connections to other graph polynomials. Furthermore, some more general domination-related polynomials are defined in this thesis, which shows some possible directions for further research. / In dieser Dissertation werden verschiedene, zum Dominationspolynom verwandte, Graphenpolynome, wie das zusammenhängende Dominationspolynom, das unabhängige Dominationspolynom und das totale Dominationspolynom, untersucht. Es werden grundlegende Eigenschaften erforscht und Sätze für die Berechnung dieser Polynome in speziellen Graphenklassen bewiesen. Weiterhin werden Ergebnisse für die Berechnung in Produktgraphen und verschiedene Repräsentationen für diese Graphenpolynome gezeigt. Ein Fokus der Dissertation liegt auf der Verallgemeinerung der verschiedenen Dominationspolynome. In diesem Zusammenhang wird das trivariate Dominationspolynom definiert. Außerdem werden Ergebnisse für das Bipartitionspolynom bewiesen. Diese beiden Polynome haben viele interessante Eigenschaften und Beziehungen zu anderen Graphenpolynomen. Darüber hinaus werden weitere multivariate Graphenpolynome definiert, die eine mögliche Richtung für weitere Forschung auf diesem Gebiet aufzeigen.