# M-Harmonious Labeling

# Abstract

Based on harmonious labelings, we introduce a new graph labeling: A multiplicative harmonious (M-harmonious) labeling is an injective function f:V(G) → U(n) that induces an injective function f′:E(G) → U(n) defined by f′(uv) = f(u)∗f(v), where U(n) is the group under modular multiplication made of integers relatively prime to n and * is the group operation. We present results in this area, including a comparison to harmonious labelings, a description of the Java program we wrote to find multiplicative harmonious labelings for a given graph, and an investigation of graphs that fail to be multiplicative harmonious.

# Original Work

While developing the paper we intend to submit for publication, this section is left as a placeholder.

# Availability of Code

The code is currently available on its GitHub project page here. The project can be downloaded and ran on any machine with the Java runtime environment.

# Using the Code

While developing the paper we intend to submit for publication, this section is left as a placeholder.

# 53rd SE Conference Boca Raton

My collaborator and I were sponsored by YSU to give a presentation at the 53rd Southeastern International Conference on Combinatorics, Graph Theory & Computing. We presented the labeling schema, as well as the LabelingFInder application.

# Future Work

Since the program is currently without a graphical interface, one could be constructed so as to make the code more easily used by researchers.

The code is currently written in Java, but again increase the approachability of the code, it can be refactored into Python or some other language that is more familiar to mathematics research.

Various results regarding M-harmonious graphs have yet to be discovered