Chromatic polynomials and order ideals of monomials

Abstract

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=1 n−1 t i(1−x) i ), where t i is the number of spanning trees with external activity 0 and internal activity i. Moreover, it is known (via commutative ring theory) that t i 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.