The chromatic polynomial of a graph

Abstract

Firstly we express the chromatic polynomials of some graphs in tree form. We then Study a special product that comes natural and is useful in the calculation of some Chromatic polynomials. Next we use the tree form to study the chromatic polynomial Of a graph obtained from a forest (tree) by "blowing up" or "replacing" the vertices Of the forest (tree) by a graph. Then we give explicit expressions, in terms of induced Subgraphs, for the first five coefficients of the chromatic polynomial of a connected Graph. In the case of higher order graphs we develop some useful computational Techniques to obtain some higher order coefficients. In the process we obtain some Useful combinatorial identities, some of which are new. We discuss in detail the Application of these combinatorial identities to some families of graphs. We also discuss Pairs of graphs that are chromatically equivalent and graph that are chromatically Unique with special emphasis on wheels. In conclusion,