  We establish the relationship between volumes of flow polytopes associated to
signed graphs and the Kostant partition function. A special case of this
relationship, namely, when the graphs are signless, has been studied in detail
by Baldoni and Vergne using techniques of residues. In contrast with their
approach, we provide entirely combinatorial proofs inspired by the work of
Postnikov and Stanley on flow polytopes. As a fascinating special family of
flow polytopes, we study the Chan-Robbins-Yuen polytopes. Motivated by the
beautiful volume formula $\prod_{k=1}^{n-2} Cat(k)$ for the type $A_n$ version,
where $Cat(k)$ is the $k$th Catalan number, we introduce type $C_{n+1}$ and
$D_{n+1}$ Chan-Robbins-Yuen polytopes along with intriguing conjectures
pertaining to their properties.
