## Abstract

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_{Γ,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_{Γ,T} modules is isomorphic to the Euclidean lattice Z^{E(Γ)}, and we describe the sublattices of integer cuts and integer flows on Γ in terms of the representation theory of A_{Γ,T}. The grading on A_{Γ,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 (Γ_{1},T_{1}) is isomorphic to the q-flow lattice of (Γ_{2},T_{2}) if and only if there is a cycle preserving bijection from the edges of Γ_{1} to the edges of Γ_{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.

Original language | English |
---|---|

Article number | 105534 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 185 |

DOIs | |

Publication status | Published - Jan 2022 |