Explicit computation of certain Arakelov-Green functions

Arakelov-Green functions defined on metrized graphs have an important role in relating arithmetical problems on algebraic curves to graph-theoretical problems. In this paper, we clarify the combinatorial interpretation of certain Arakelov-Green functions by using electric circuit theory. The formulas we give clearly show that such functions are piecewisely defined, and each piece is a linear or quadratic function on each pair of edges of metrized graphs. These formulas lead to an efficient algorithm for explicit computation of Arakelov-Green functions.