  In this article we introduce the notion of weak harmonic labeling of a graph,
a generalization of the concept of harmonic labeling defined recently by
Benjamini et al. that allows extension to finite graphs and graphs with leaves.
We present various families of examples and provide several constructions that
extend a given weak harmonic labeling to larger graphs. In particular, we use
finite weak models to produce new examples of (strong) harmonic labelings. As a
main result, we provide a characterization of weakly labeled graphs in terms of
harmonic subsets of the integers and use it to compute every such graphs of up
to ten vertices. In particular, we characterize harmonically labeled graphs as
defined by Benjamini et al. We further extend the definitions and main results
to the case of multigraphs and total labelings.
