Chromatic polynomials and order ideals of monomials Original Research Article

One expansion of the chromatic polynomial π(G,x) of a graph G relies on spanning trees of a graph. In fact, for a connected graph G of order n, one can express π(G,x)=(−1)n−1x∑i=1n−1ti(1−x)i), where ti is the number of spanning trees with external activity 0 and internal activity i. Moreover, it is known (via commutative ring theory) that ti arises as the number of monomials of degree n − i − 1 in a set of monomials closed under division. We describe here how to explicitly carry out this construction algebraically. We also apply this viewpoint to prove a new bound for the roots of chromatic polynomials.