Chromatic Equivalent Graphs Of Two Kinds Of Graphs

In 1978,Chao andWhitehead defined a graph to be chromatically unique if no othergraphs share its chromatic polynomial.It is well-known that the chromatic polynomialof a graph is one of the basic tools for studying the chromaticity of graphs, whichis denoted by p(G,λ) for graph G.Two graphs G and H are said to be chromaticallyequivalent,denoted by G～H, if p(G,λ) = p(H,λ). A graph G is called chromaticallyunique if G (=|～) H, whenever G～H.Since then various families results of this regionhave been found.The notion of adjoint polynomial of first introduced,and in virtue of it, succeeded instudy of the chromaticity of graphs Liu ([2], [18]) in 1987, which was used to investigatethe chromaticity of a graph from its complement. Denoted by h(G, x) the adjointpolynomial of a graph G , two graphs G and H are said to be adjoint equivalent,denotedby G～h H,if h(G, x) = h(H, x).A graph G is called to be adjointly unique if H (=|～) G,whenever H～h G. In fact, two graphs G and H are said to be adjointly equivalentif and only if their complement (G|-) and (H|-) are said to be chromatically equivalent, andgraph G is adjointly unique if and only if its complement (G|-) is chromatically unique.The related contents could be found in ([2], [3], [5], [6], [10], [12], [18], [20]～[25]).In this paper, I mainly by use of the properties of adjoint polynomial, such as: divisibility,characters and minimum real roots and so on. Charactered the adjointequivalent graphs of two graphs, then obtained the chromatic equivalent graphs andthe condition of chromatic uniqueness of their complement. The main contents of eachchapter of this thesis are as follows:Chapter 1: The preliminary knowledge of adjoint polynomials is introduced;Chapter 2: The chromatic equivalent graphs of ((A_n∪kD_m)|-)(m≥5) are charactered;Chapter 3: The chromatic equivalent graphs of ((B_n∪kD_m)|-)(m≥5) are charactered;...