Computations Of Chromatic Polynomials Of Graphs

The computations of chromatic polynomials are the most essential problems that arise often in various hot branches of the roots of chromatic polynomials,the coefficients of chromatic polynomials and the chromaticity of graphs.The inclusion-exclusion principle is a classical enumerative method in combinatorics.The main job of this paper is to compute the chromatic polynomials by using inclusion-exclusion principle,which may be summarized as follows.The first chapter introduces the background about the origin,development and the computation method of the chromatic polynomials of graphs.The second chapter introduces the basic concepts of graphs and the chromatic poly-nomials of graphs.In the third chapter,it makes the question more simplified to discuss two different conditions whether containing cycles during computing the chromatic polynomials by using inclusion-exclusion principle.As applications,the chromatic polynomials for the one-cycle graphs and three kinds of double-cycles graphs are proved respectively.In the fourth chapter,we obtain the absolute sum of chromatic polynomials for double-cycles and n-cycles with a common vertex.Furthermore,the upper bounds on the absolute sum of chromatic polynomials are improved efficiently.At last,it can prove the Stirling number of the second kind by using chromatic polynomials of star graph.