Commutative distance degree-regular graphs

Abstract

This paper introduces and studies a family of graphs characterized by a commutative structure among their distance matrices, with a focus on those exhibiting distance-based regularity via constant row sums. This notion extends classical degree-regularity to a broader context, and, in particular, presents a new generalization of distance-regular graphs. We provide characterizations of this class, examine their relation to classical notions of regularity, and present several examples and constructions that highlight their behavior.