We provide a homological algebraic realization of the lattices of integer cuts and integer flows of graphs. To a finite 2-edge-connected graph Γ with a spanning tree T, we associate a finite dimensional Koszul algebra $A_{\mathrm{\Gamma },T}$. Under the construction, planar dual graphs with dual spanning trees are associated Koszul dual algebras. The Grothendieck group of the category of finitely-generated $A_{\mathrm{\Gamma },T}$ modules is isomorphic to the Euclidean lattice ${\mathbb{Z}}^{E(\mathrm{\Gamma })}$, and we describe the sublattices of integer cuts and integer flows on Γ in terms of the representation theory of $A_{\mathrm{\Gamma },T}$. The grading on $A_{\mathrm{\Gamma },T}$ gives rise to q-analogs of the lattices of integer cuts and flows; these q-lattices depend non-trivially on the choice of spanning tree. We give a q-analog of the matrix-tree theorem, and prove that the q-flow lattice of $({\mathrm{\Gamma }}_1,T_1)$ is isomorphic to the q-flow lattice of $({\mathrm{\Gamma }}_2,T_2)$ if and only if there is a cycle preserving bijection from the edges of ${\mathrm{\Gamma }}_1$ to the edges of ${\mathrm{\Gamma }}_2$ taking the spanning tree $T_1$ to the spanning tree $T_2$. This gives a q-analog of a classical theorem of Caporaso-Viviani and Su-Wagner.

我们提供了图的整数割和整数流的格的同调代数实现。对于具有生成树T的有限 2 边连通图 Γ ，我们将有限维 Koszul 代数关联起来${\mathrm{一种}}_{\mathrm{\Gamma },吨}$. 在构造下，具有双生成树的平面对偶图与 Koszul 对偶代数相关联。有限生成范畴的格罗腾迪克群${\mathrm{一种}}_{\mathrm{\Gamma },吨}$ 模块与欧几里得格同构 ${\mathbb{Z}}^{乙(\mathrm{\Gamma })}$，我们根据 Γ 的表示理论描述了整数割和整数流的子格 ${\mathrm{一种}}_{\mathrm{\Gamma },吨}$. 评分在${\mathrm{一种}}_{\mathrm{\Gamma },吨}$产生整数切割和流动的格子的q类比；这些q格主要取决于生成树的选择。我们给出矩阵树定理的q类比，并证明q流格$({\mathrm{\Gamma }}_1,{吨}_1)$与q流晶格同构$({\mathrm{\Gamma }}_2,{吨}_2)$ 当且仅当存在来自边缘的循环保持双射 ${\mathrm{\Gamma }}_1$ 到边缘 ${\mathrm{\Gamma }}_2$ 取生成树 ${吨}_1$ 到生成树 ${吨}_2$. 这给出了Caporaso-Viviani 和 Su-Wagner 经典定理的q类比。