Generating function and root properties for chromatic polynomials

Abstract

The chromatic polynomial P(G,t) related to graphs is a major area of algebraic graph theory. These polynomials allow graphs to be extracted using algebraic techniques and contain important information about their structure and properties. In particular, the coefficients, roots, and derivatives of graphing polynomials at certain points and their values are frequently interpreted in important ways. A graph with vertices V and a chromatic polynomial chi_{G}(n) is called G. In the paper, we discover some properties by defining h_{G}(z) as the generating function of the chromatic polynomial of the graph along with properties of the roots. A real number t is called a chromatic root of the generating function for the chromatic polynomial of a graph G if h(G, t) = 0. Since the chromatic polynomial enumerates the appropriate colors of a graph, it is natural to see how color roots behave in light of the assumptions related to them.