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Combinatorial Iterated Integrals and the Harmonic Volume of Graphs

Let $Γ$ be a connected bridgeless metric graph, and fix a point $v$ of $Γ$. We define combinatorial iterated integrals on $Γ$ along closed paths at $v$, a unipotent generalization of the usual cycle pairing and the combinatorial analogue of Chen's iterated integrals on Riemann surfaces. These descend to a bilinear pairing between the group algebra of the fundamental group of $Γ$ at $v$ and the tensor algebra on the first homology of $Γ$, $\int\colon \mathbf{Z}π_1(Γ,v) \times T\mathrm{H}_1(Γ,\mathbf{R}) \to \mathbf{R}$. We show that this pairing on the two-step unipotent quotient of the group algebra allows one to recover the base-point $v$ up to well-understood finite ambiguity. We encode the data of this structure as the combinatorial harmonic volume which is valued in the tropical intermediate Jacobian. We also give a potential-theoretic characterization for hyperelliptiicity for graphs.