The Adjoint Uniqueness And Chromaticity Of Two Kinds Of ψ-Graphs

The notinon of chromatically unique graphs was first introduced and studied by Chao and Whitehead([2])in 1978.The chromatic polynomials of a graph is mainly applied for studying the chromaticity of graphs.By P(G,λ) is denoted chromatic polynomial of a graph G.Two graphs G and H are said to be chromatically equivalent,denoted by G～H,if P(G,λ) = P(H,λ).A graph G is called chromatically unique if H≌G,whenever H～G. The related papers could be found in the referencess[7,8].In 1987,the adjoint polynomial of a graph was first introduced by professor Ruying Liu in[12],which was a new method to study chromaticity of a graph from the angle of its complement.Denoted by h(G,x) the adjoint polynomials of a graph G,Two graphs G and H are said to be adjointly equivalent,denoted by G(?)H,if h(G,x) = h(H,x).A graph G is called to be adjointly unique if H≌G,whenever H(?)G.The relations of the chromatic polynomial of a graph and its adjoint polynomials are that:graphs G and H are adjoint equivalent if and only if their complement(?) and(?) are chromatically equivalent,and graph G is adjoint uniqueness if and only if its complement(?) is chromatic uniqueness.Many results in this filed could be seen in the referencess([3]-[6],[9]-[17],[19]-[22]).In this thesis,the adjoint uniqueness and chromaticity of two kinds of graphs are chiefly studied by means of the properties of adjoint polynomials,including divisibility,characters and minimum real roots and so on.The main contents of each chapter of the thesis are as follows:Chapter 1:Introduction of the preliminary knowledges of adjoint polynomials;Chapter 2:Preliminaries of adjoint polynomial are presented;Chapter 3:Studying the adjoint uniqueness ofψ_n~4(1,n-6);Chapter 4:Studying the adjoint uniqueness ofψ_n~3(n-3,1).