Flow polytopes for extensions of bipartite graphs

TL;DR

This paper connects the geometry of flow polytopes to classical graph combinatorics by associating to a bipartite DAG $H$ an extension graph $G(H)$ and an almost-degree-whiskered graph $W(H)$. It proves that the normalized volume of the unit-flow polytope $\mathcal{F}_1(G(H))$ equals the number of matchings in $W(H)$, and that the Ehrhart $h^*$-polynomial $h^*(\mathcal{F}_1(G(H));z)$ equals the unsigned matching polynomial $\mu(W(H);z)$. The results rely on the Danilov–Karzanov–Koshevoy unimodular triangulations with a canonical bipartite framing, together with a refined bijection between clique structures in $G(H)$ and matchings in $W(H)$. As a consequence, $h^*$ is real-rooted and unimodal, linking polyhedral invariants to well-studied graph polynomials and offering a combinatorial handle on flow polytope volumes.

Abstract

The space of unit flows on a finite acyclic directed graph is a lattice polytope called the flow polytope of the graph. Given a bipartite graph $G$ with minimum degree at least two, we construct two associated acyclic directed graphs: the extension of $G$ and the almost-degree-whiskered graph of $G$. We prove that the normalized volume of the flow polytope for the extension of $G$ is equal to the number of matchings in the almost-degree-whiskered graph of $G$. Further, we refine this result by proving that the Ehrhart $h^*$-polynomial of the flow polytope for the extension of $G$ is equal to the unsigned matching polynomial of the almost-degree-whiskered graph of $G$.

Figures (6)

• Figure 1: The graph extension for $K_{3,2}$.
• Figure 2: The $\mathcal{H}$-corona of $G$ described in Example \ref{['ex: H-corona']}.
• Figure 3: An example of a framed $G(K_{3,2})$. Note that five of the edges have been labeled by their $\alpha$, $\beta$, $\gamma$ notation. Note also that the framing labels agree with the orderings induced by the canonical bipartite framing.
• Figure 4: The routes from Example \ref{['ex:cliquemap']}. For clarity, edges and inner vertices have been duplicated when multiple routes pass through them. Thus, for example, edge $\beta_{3,5}$ is shown twice, once as a solid line and once as a dashed line.
• Figure 5: Two matchings $\psi(\mathbf{a})$ and $\psi(\mathbf{a}')$ of $W(K_{3,2})$.

Theorems & Definitions (53)

• Definition 1.1
• Definition 1.2
• Example 1.3
• Definition 1.4
• Example 1.5
• Definition 1.6
• Theorem 1.7
• proof
• Theorem 1.8
• Corollary 1.9