The Interaction Of Bracket Polynomial And Dichromatic Polynomial

Given a plane graph G=(V,E)，where V and E denote the vertex set and edge set of G,respectively．All graphs considered in this paper are unorientable and simple．For alternating diagram of an alternating knots, there are four regions. The two regionsrotating the over-crossing line to under-crossing line by counterclockwise denote the shadedparts on the alternating diagram. K (G)denotes the dual alternating knot of the planegraph G．Let [K (G)]， Z_G(q,v)，T_G (x,y)denote the bracket polynomial of K (G)，thedichromatic polynomial and the Tutte polynomial of G, respectively．In this paper, we discuss the relationships of [K (G)]， Z_G(q,v)and T_G (x,y)byusing their properties. We give a new and simple computational approach and some specialcharacters of the bracket polynomial of alternating knots．Using the bracket polynomial andthe Tutte polynomial, we discuss the number of ways to vertex-color for the graph G．Sowe give a new study way between graph theory and knot theory．Universe graphs are in one-to-one correspondence with planar graphs．A universe graphcorresponds two alternating knots，one is the mirror image of the other．The bracketpolynomials of two alternating knots have certain correlation transformation．We give a newdefinition to [K (G)] and Z_G(q,v)satisfying a relationship，so that some alternating knotsare in one-to-one correspondence with planar graphs．In this paper，we’ll try to give sometheorems and propositions through mathematical induction and the analogies proof method，so deepen the mutual understanding of the connection of the three polynomials．we give asimplification algorithm of bracket polynomial．In the end of this paper,we discuss therelated application of bracket polynomials and the Tutte polynomial in the theory ofvertex-coloring of the planar graph G．...