The Adjoint Polynomial And The Chromaticity Of Several Kinds Of Graphs

In 1978,Chao and Whitehead([2]) defined a graph to be chromatically unique if onother graphs share its chromatic polynomial.The chromaticity of graphs is denotedby P(G,λ) for graph G.Two graphs G and H are said to be chromatically equiva-lent,denoted by G～H.So far,a lot of classes graphs of chromatic uniqueness havebeen founded successively, refering to ([2]～[4],[7],[8]).The notion of adjoint polynomial of first introduce,and in virtue of it,succeededin study of chromaticity of graphs Liu[25] in 1987,which was used to investigatethe chromaticity of a graph from its complement.By h(G,x) we denote the adjointpolynomial of G.If h(G,x) = h(H,x),two graphs G and H are said to be adjointlyequivalent,denoted by G h H.A graph G is called adjointly unique if G =～HwheneverG h H.Adjiont polynomial of a graph likewise is one of basic tools forstudying the chromaticity of graphs.As a matter of fact,two graphs G and H aresaid to be adjointly equivalent if and only if theirs complement graph G and Hsaid to be chromatically equivalent and graph G is adjointly unique if and only ifits complement G is chromatically unique.More related results can be founded in([5],[9]～[11],[14],[15],[23]～[35]).In the thesis,we discuss the chromaticity of the following graphs mainly via theproperties of the adjoint polynomials,including minimum adjoint roots, character,divisibility, especial complements,categorizing discussion method and so on.We prove the graphψn3(n ? 5,3) is chromatically unique if and only if n≥10;We prove two graphs (?) are adjoint equiva-lent,when n≥7,n = 9,12;We prove the order of minimal roof of (?)...