Combinatorial iterated integrals and the harmonic volume of graphs

Let Γ be a connected bridgeless metric graph, and fix point v of Γ. We define combinatorial iterated integrals on along closed paths at v, unipotent generalization the usual cycle pairing analogue Chen's Riemann surfaces. These descend to bilinear between group algebra fundamental tensor first homology Γ, ∫:Zπ1(Γ,v)×TH1(Γ,R)→R. show that this two-step quotient allows one recover base-point up well-understood finite ambiguity. encode data structure as harmonic volume which is valued in tropical intermediate Jacobian. also give potential-theoretic characterization for hyperellipticity graphs.