Broken circuit complexes of series-parallel networks

Let (h0,h1,…,hs) with hs≠0 be the h-vector of the broken circuit complex of a series-parallel network M. Let G be a graph whose cycle matroid is M. We give a formula for the difference hs−1−h1 in terms of an ear decomposition of G. A number of applications of this formula are provided, including several bounds for hs−1−h1, a characterization of outerplanar graphs, and a solution to a conjecture on A-graphs posed by Fenton. We also prove that hs−2≥h2 when s≥4.