  In this paper, we extend the classical arithmetic defined over the set of
natural numbers N, to the set of all finite directed connected multigraphs
having a pair of distinct distinguished vertices. Specifically, we introduce a
model F on the set of such graphs, and provide an interpretation of the
language of arithmetic L={0,1,<=,+,x} inside F. The resulting model exhibits
the property that the standard model on N embeds in F as a submodel, with the
directed path of length n playing the role of the standard integer n. We will
compare the theory of the larger structure F with classical arithmetic
statements that hold in N. For example, we explore the extent to which F enjoys
properties like the associativity and commutativity of + and x, distributivity,
cancellation and order laws, and decomposition into irreducibles.
