The basis digraphs of p-schemes

TL;DR

This paper characterizes p-schemes through their basis digraphs, establishing that association schemes with bipartite basis graphs are precisely 2-schemes, thus linking graph structure to algebraic properties.

Contribution

It provides a new characterization of p-schemes using basis digraphs, extending understanding of the structure of association schemes.

Findings

Association schemes with bipartite basis graphs are exactly 2-schemes.

p-schemes can be characterized by their basis digraphs for any prime p.

The result connects graph bipartiteness with algebraic scheme properties.

Abstract

It is proved that association schemes with bipartite basis graphs are exactly 2-schemes. This result follows from a characterization of p-schemes for an arbitrary prime p in terms of basis digraphs.