Chromatic M-polynomial of Mobius Function Graphs of Product Groups

Abstract

Objective: To investigate the chromatic M-polynomials of the Mobius function graph of product groups, specifically focusing on the direct product of finite groups. Methods: We used algebraic and number theoretical techniques to construct a Mobius function graph of product groups and derived the chromatic M-polynomials using graph theoretic methods. Findings: Chromatic M polynomials of Mobius function graphs are introduced with a focus on deriving their significant properties and providing a practical approach for computation. Novelty: This research introduces a novel approach to studying the chromatic M-polynomials of the Mobius function graph of product groups, providing insights into the interplay between group theory, graph theory, and number theory. Our findings contribute connection between chromatic M polynomials and coloring problems, followed by a discussion and understanding of graph polynomials. Keywords: Mobius function graph of groups, Chromatic number, Chromatic M-polynomials, Product group, Dihedral Groups