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.