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 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2020
 arXiv:
 arXiv:2011.14063
 Bibcode:
 2020arXiv201114063L
 Keywords:

 Mathematics  Combinatorics;
 05C78;
 05C63;
 05C75;
 05C85
 EPrint:
 21 pages, 16 figures