中文摘要：

最近Klaus Dohmen等人提出新的双变量色多项式概念,对此,本文提出一个一般性的减边公式.通过反复运用该公式,可以方便求得任何简单图的双变量色多项式.由此减边公式,研究了一些特殊图和多分支图的双变量色多项式公式.本文还研究了由互不相连的多个子图都与某个顶点相连而成的图的双变量色多项式计算的删点公式以及简单图的双变量色多项式系数和问题.进而,本文提出一个新概念—正则树.利用这个减边公式,研究了正则树的双变量色多项式计算公式和一些性质,以及正则树整子图的双变量色多项式公式及其有关性质.

英文摘要：

Recently, Klaus Dohmen et al proposed a notion of new two-variable chromatic polynomials. In order to further study the related properties of this notion, we have Carried out theoretical derivation and computer verified, then we have achieved a general compute formula for the two-variable chromatic polynomials of any graph, calling Reduction Edge Formula, we give it a forthright proof. By using the formula repeatedly, people can get any graph＇s two-variable chromatic polynomials conveniently. Based on this formula, we got the two-variable chromatic polynomial compute formula of some special graphs and disconnected multi-branch graph. Also, we got a Delete Vertex Formula of a graph which contains disconnected multi-partite as its subgraphs, and a vertex linked to each subgraph. We study the sum of coefficients to two-variable chromatic polynomials and we get a important conclusion for it. Then we propose a new notion of regular tree, and we study its new two-variable chromatic polynomial compute formula and many other properties. Later, we propose integral subgraph of a regular tree , and we study the new two-variable chromatic polynomial of it, also we got its compute formula and many other important properties.