The Unique Of Chromaticity And The Adjoint Polynomial Of Some Classes Of Graphs

In 1978, Chao and Whitehead ([2]) defined a graph to be chromatically unique if on other graphs share its chromatic polynomial. It is well-know that the chromaticpolynomial of a graph is one of the basic tools for studying the chromaticity of graphs, which is denoted by P(G,λ) for graph G. Two graphs G and H are said to be chromatically equivalent, denoted by G - H, if P(H,λ) = P(G,λ). A graph G is called chromatically unique if H≌G whenever H - G. So far, many classes graphs of chromatic uniqueness have been appeared successively refering to ([2]-[4],[9],[10]).The notion of adjoint polynomials of first introduced, and in virtue of it, suceeded in study of the chromaticity of graphs Liu [11] in 1987, which was used to investigate the chromaticity of a graph from its complement. By h(G, x) we denote the adjoint polynomial of graph G. Similarly, two graphs G and H are said to be adjointly equivalent, denoted by G -~h H, if P(H,λ) = P(G,λ). A graph G is called adjointly unique if H≌G whenever H -~h G. Adjoint polynomial of a graph likewise is one of the basic tools for studying the chromaticity of graphs. In fact, Two graphs G and H are said to be adjointly equivalent if and only if theirs complement (?) and (?) are said to be chromatically equivalent and graph G is adjointly unique if and only if its complement (?) is chromatically unique. The related paper could be found in([5],[6],[8],[11]-[16],[18]-[28]).In this thesis, we inverstigate the chromaticity of the following graphs mainly utilized the properties of the adjoint polynomials , including divisibility, minimum adjoint roots, especial complement, character and so on.We prove that graph (?) is chromatically unique if and only if n≠8, 11;We obtain that (?) is unique chromaticity if and only if n≥11;We have that (?) is chromatically unique if and only if n≠8, 9.