  We prove results describing the structure of a Chow ring associated to a
product of graphs, which arises from the Gross-Schoen desingularization of a
product of regular proper semi-stable curves over discrete valuation rings. By
the works of Johannes Kolb and Shou-Wu Zhang, this ring controls the behavior
of the non-Archimedean height pairing on products of smooth proper curves over
non-Archimedean fields. We provide a complete description of the degree map,
and prove vanishing results affirming a conjecture of Kolb, which, combined
with his work, leads to an analytic formula for the arithmetic intersection
number of adelic metrized line bundles on products of curves over complete
discretely valued fields.

Algebraic Geometry;Combinatorics;Number Theory