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arXiv · 2509.26445

Flow polytopes for extensions of bipartite graphs

Benjamin Braun·Kaitlin Bruegge·Robert Davis·Derek Hanely
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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 $G$.

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Benjamin Braun, Kaitlin Bruegge, Robert Davis, Derek Hanely. 2025-09-30. Flow polytopes for extensions of bipartite graphs. https://arxiv.org/abs/2509.26445

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