The Factorization of Adjoint Polynomials of E^G（i）-class Graphs and Chromatically Equivalence Analysis

Let Sn be the star with n vertices, and let G be any connected graph with p vertices. We denote by Eτp＋（r-1）^G（i） the graph obtained from Sr and rG by coinciding the i-th vertex of G with the vertex of degree r - 1 of S,, while the i-th vertex of each component of （r - 1）G be adjacented to r - 1 vertices of degree 1 of St, respectively. By applying the properties of adjoint polynomials, We prove that factorization theorem of adjoint polynomials of kinds of graphs Eτp＋（r-1）^G（i）∪（r - 1）K1 （1 ≤i≤p）. Furthermore, we obtain structure characteristics of chromatically equivalent graphs of their complements.

On Chromatically Equivalence of a Class of Graphs

A class of new graphs is defined. A sufficient condition for pairs of these graphs to be chromatically equivalent is proven.

Graph Description by the Chromatic Polynomials

By means of the chromatic polynomials, this paper provided a necessary and sufficient condition for the graph G being a mono-cycle graph（the Theorem 1）, a first class hi-cycle graph and a second class bicycle graph（the Theorem 2）, respectively.

An Approach to Chromaticity of K_4-homeomorphs

We introduce a new method for studying the chromaticity of K4-homeomorphs,by which some new results are obtained.

Chromatic Uniqueness of a Family of K_4-Homeomorphs

It is shown that K 4(i,j,l,k,m,n) is chromatically unique if three numbers among i,j,l,k,m,n have the same value and the other three numbers are not equal but larger than that value.

Graphs with the Same Color-partition as the Complement of lK_1∪(∪C_(u_i))

In this paper, using the properties of chromatic polynomial, we discuss the color-partition of the complement of lK 1∪(∪C u i),and characterize the graph with the same color-partition as the class graph under u i≠4k+2.

Graphs with the Same Color-partition as the Complement of lK1∪(∪Cui)

In this paper, using the properties of chromatic polynomial, we discuss the color-partition of the complement of lK 1∪(∪C u i),and characterize the graph with the same color-partition as the class graph under u i≠4k+2.

The Relation on the Coefficients and Roots of Adjoint Polynomial and Its Application

The parameter R（G） is the function about the front three coeffcients of the adjoint polynomial of graph G. In the paper, the range of R（G） is given when β（G） 〈 β（Dn）, where β（G） is the minimum root of the adjoint polynomial of graph G and the chromatically equivalent classification of tDn is completely depicted.Furthermore, a sufficient and necessary condition for the class of graphs to be chromatically unique is obtained.