The Chromaticity Of Several Classes Of Graphs And The Fifth Coefficient Of Adjoint Polynormial

In 1978, Chao and Whitney [22] defined a graph to be chromatically unique if no other graphs share its chromatic polynomial. They gave several families of chromatically unique graphs. Since then many researchers have been study-ing chromatic uniqueness and chromatic equivalence of graphs. The question on chromatic equivalence and uniqueness is called the chromaticity problem of graphs.Two graphs are defined to be adjointly equivalent if their complements are chromatically equivalent. Hence the goal of determining chromatic equivalence class for a given graph can be realized by determining adjoint equivalence class of its complement. Thus if the size of a given graph is very large, it may be easier to study adjoint equivalence class rather than chromatic equivalence class. The adjoint polynomial of a graph is a useful tool for this study, which was introduced by Liu in [6].In this thesis, the main aim is to study the chromaticity of some classes of graphs. In Chapter 2, the adjoint equivalence class of Bn-8,1,4 and Bn-9,1,5 are determined by the fourth character R4(G). In Chapter 3, a new invariant of graph G, which is the fifth character R5(G), is given in this paper. Using this invariant and the properties of the adjoint polynomials, we completely determine the adjoint equivalence class ofψn3(n—3,1) andζn1. Finally, the expression of the fifth coefficient of adjoint polynomial is given in Chapter 4.