Cayley graphs from classes of nilpotent rings

Abstract

This article introduces the construction of Cayley graphs arising from the additive groups of finite nilpotent rings, where the connection set is chosen to reflect the nilpotent structure. We investigate the interplay between ring-theoretic properties—such as nilpotency index, characteristic, and ideal structure—and graph-theoretic parameters including regularity, diameter, girth, and spectral features. Several families of nilpotent rings (e.g., rings of strictly upper triangular matrices, truncated polynomial rings, and rings with trivial multiplication) are examined as concrete sources of Cayley graphs with controlled combinatorial behaviour. The main results establish bounds on diameter and girth in terms of the nilpotency index, characterise the adjacency spectrum for abelian cases, and highlight connections to strongly regular graphs. This work aims to bridge algebraic ring theory and algebraic graph theory, offering a new class of structured graphs for potential applications in network design and coding theory.