Bridging group theory and graph theory through commutativity degrees

The interplay between algebraic structures and graph theory has emerged as a dynamic and fruitful area of research in modern mathematics. By establishing connections between graphs and groups, mathematicians have developed innovative approaches to visualize and analyze the intrinsic properties of groups using graph-theoretic tools. These connections enhance our understanding of group theory and provide new perspectives on how graph theory can be effectively applied within algebraic frameworks.

The origins of this interdisciplinary relationship can be traced back to the pioneering work of Arthur Cayley in 1878, who introduced what are now famously known as Cayley graphs. These graphs serve as visual representations of groups, where each vertex corresponds to a group element, and edges reflect the relationships defined by a generating set. Cayley’s groundbreaking idea laid the foundation for a vast and ongoing exploration of the rich interactions between group theory and graph theory, which continues to inspire contemporary research across various mathematical domains.

In this study, we contribute to this line of research by investigating different notions of commutativity degrees in finite groups and exploring their algebraic properties. Building on these findings, we construct new classes of graphs associated with finite groups and examine their structural characteristics. We further illustrate our results by considering several examples of special types of groups, offering fresh insights into the mutual influence of algebraic and graph-theoretic perspectives.