  Let $\Gamma$ be a connected bridgeless metric graph, and fix a point $v$ of
$\Gamma$. We define combinatorial iterated integrals on $\Gamma$ 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 $\Gamma$ at $v$ and the tensor algebra on the first homology of
$\Gamma$, $\int\colon \mathbf{Z}\pi_1(\Gamma,v) \times
T\mathrm{H}_1(\Gamma,\mathbf{R}) \to \mathbf{R}$. 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.
