Weak harmonic labeling of graphs and multigraphs

TL;DR

This paper introduces weak harmonic labeling for graphs, extending harmonic labeling concepts to finite graphs with leaves, and provides characterizations, constructions, and classifications of such labeled graphs.

Contribution

It generalizes harmonic labeling to weak harmonic labeling, extends the concept to multigraphs, and offers a complete characterization of small weakly labeled graphs.

Findings

Characterization of weakly labeled graphs via harmonic subsets of integers

Complete classification of weakly labeled graphs up to ten vertices

Extension of harmonic labeling concepts to multigraphs and total labelings

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.