Types Adjoint Polynomial Nature

In 1978,Chao and Whitehead([2]) defined a graph to be chromatically unique if on other graphs share its chromatic polynomial. The chromaticity of graphs is denoted by 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 have been founded successively, refering to ([2]～[4],[6],[7]).The notion of adjoint polynomial of first introduce,and in virtue of it, succeeded in study of chromaticity of graphs Liu[15] in 1987, which was used to investigate the chromaticity of a graph from its complement. By h(G, x) we denote the adjoint polyno-mial of G.If h(G, x)= h(H, x), two graphs G and H are said to be adjointly equivalent, denoted by G～h H.A graph G is called adjointly unique if G≌H wheneverG～h H. Adjiont polyno-mial of a graph likewise is one of basic tools for studying the chromaticity of graphs. As a matter of fact, two graphs G and H are said to be adjointly equivalent if and only if theirs complement graph G and H said to be chromatically equivalent and graph G is adjointly unique if and only if its complement G is chromatically unique. More related results can be founded in ([5],[8]～[15].In the thesis,there are five chapters.The main contents of each chapter of this thesis axe as follows:Chapter 1:Introduction of the preliminary knowledge of adjoint polynomials; Chapter 2:Preliminaries of adjoint polynomials are presented; Chapter 3:The recurrence relation of Fn with Fn,Dn and Pn; Chapter 4:Discussion on the minimum real roots of the adjoint polynomials of a class of connected graphs with character-2 and cyclomatic number 1 were characterized,in which the root extremal graphs were given. Chapter 5:Discussion on the minimum real roots of the adjoint polynomials of connected graphs in which two points with degree 2 of a triangle are identified with Fn,D'n,Dn,Cn or Path, and some new results are obtained.