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 the Minimum Real Roots of the Adjoint Polynomials of Graphs

In this paper, we are concerned with the minimum real root of the adjoint polynomial of the connected graph G with cut-vertex u, in which G - u contains paths, circles or Dn components. Here Dn is the graph obtained from K3 and path Pn-2 by identifying a vertex of K3 with an end-vertex of Pn-2. Some relevant ordering relations are obtained. This extends several previous results on the minimum roots of the adjoint polynomials of graphs.

THE ROOTS OF σ-POLYNOMIALS

Let G be a connected graph. We denote by σ(G,x) and δ(G) respectively the σ-polynomial and the edge-density of G,where δ(G)=|E(G)||V(G)|2. If σ(G,x) has at least an unreal root,then G is said to be a σ-unreal graph.Let δ(n) be the minimum edge-density over all n vertices graphs with σ-unreal roots. In this paper,by using the theory of adjoint polynomials, a negative answer to a problem posed by Brenti et al. is given and the following results are obtained:For any positive integer a and rational number 0≤c≤1,there exists at least a graph sequence {G i} 1≤i≤a such that G i is σ-unreal and δ(G i)→c as n→∞ for all 1≤i≤a,and moreover, δ(n)→0 as n→∞.

The Roots of Adjoint Polynomial of the Graphs Contain Triangles

We denote h(G,x) as the adjoint polynomial of graph G. In [5], Ma obtained the interpolation properties of the roots of adjoint polynomial of graphs containing triangles. By the properties, we prove the non-zero root of adjoint polynomial of Dn and Fn are single multiple.

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.

Chromatic Uniqueness of the Complement of T_m(l, 4, m - 6)

In the paper, we prove that the complement of Tm(1,4,m-6)(m> 10) is chromatically unique if and only if m≠5k(k≥2).