Polynomial With Coloring The Graph

The main content of this paper is discussing graph plane division figure coloring,through discussing the zeros of graphs’ chromatic polynomial, it analyzes the minimumnumber of ways to faces-color the graph with colors so that no two adjacent faces receive thesame color. And then gives the important properties of the number of ways to color planedivision figure. The graphs’ chromatic polynomial has connection with the partition functionof Potts Model,so there are some related conclusions of the statistical mechanics in thispaper.The major research methods of the article is giving a new study channel of coloringplane division figure, then we will compute the later graph’s dichromatic polynomial, andsummarizing the coloring properties of decomposing plane before and after,and discussing thecoloring properties of sphere division figure. The overall structure of this paper is that thereview of the knowledge of graph theory and statistical mechanics,and the discussion of theleast number of ways to coloring plane division figure before and after.The first discuss thecoloring number of n regional figure;The second discuss the coloring number of generalizeddivision figure of regional figureGn;The third discuss the coloring number of the regionalfigureGn with the regional figureG_mhave a public side and their generalized divisionfigure; The forth discuss the coloring number of two of the two adjacent three regional figureG_n, G_m,Gt,and their generalized division figure; The fifth discuss the coloring number oftriangulations figure of regional figureG_n; The sixth discuss the coloring number of thetriangulations figure of simple polyhedron and sphere.The last for the Potts Model on the plane, we will apply the above analysis of partitionfunction of Potts Model. Meanwhile various thermodynamic quantities can calculate from thepartition function. For example, calculating the probability of some model and so on. So wecan get related applications of statistical mechanics.