Broken circuit complexes of series-parallel networks

Let $(h_0,h_1,\ldots,h_s)$ with $h_s\ne0$ 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 $h_{s-1}-h_1$ in terms of an ear decomposition of $G$. A number of applications of this formula are provided, including several bounds for $h_{s-1}-h_1$, a characterization of outerplanar graphs, and a solution to a conjecture on $A$-graphs posed by Fenton. We also prove that $h_{s-2}\geq h_2$ when $s\geq 4$.