Weak harmonic labeling of graphs and multigraphs

Abstract

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, Cyr, Procaccia and Tessler 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 Z and exhibit quantitative evidence of the efficiency of this method for computing all weakly labeled finite graphs as opposed to an exhaustive search calculation. 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.