The chromatic polynomials of the q-analogue of certain graph operations

Abstract

In this dissertation, we start by giving basic definitions in graph theory and some classes of graphs which are relevant in this work. We then introduce the chromatic polynomial of a graph which is the main focus of this dissertation. In addition, we discuss some properties of a graph, and the chromatic polynomials which are relevant to this work. In particular, explicit expression of chromatic polynomials of certain classes of graphs are presented. Certain graph operations and the chromatic polynomials of the resultant graphs are discussed. Finally, we introduce the 2-vertex join of a graph and we study its chromatic polynomial, in particular we give the explicit expression of the flow polynomial of the 2-vertex join of a path Pn.