Commutative Distance-Regularity: Algebraic Hierarchies and Cayley Graph Constructions

Abstract

This paper resolves several open questions regarding the algebraic and metric inheritance of distance-based regularities under graph products, with a particular focus on Commutative Distance Degree-Regular (CDDR) graphs. We characterize the behavior of CDDR, distance mean-regular (DMR), and distance degree-regular (DDR) graphs under the strong and direct (tensor) products. To address the non-closure of classical regularities under the direct product, we introduce and study a novel, broader family of graphs: \textit{Distance Walk Regular (DWR)} graphs. Furthermore, we relax the global commutativity condition by defining and structuralizing the class of $i$-CDDR graphs, providing deep characterizations within the realm of Cayley graphs. By developing systematic lifting operations, we construct infinite families of graphs with arbitrarily large order and diameter within specific structural intersections, effectively settling topological gaps in the current regularity hierarchy.