P-Polynomial and Bipartite Coherent Configurations

TL;DR

This paper introduces P-polynomial coherent configurations, explores their structure with at most two fibres, and connects them to well-known combinatorial objects like distance-biregular graphs and strongly regular designs.

Contribution

It defines P-polynomial coherent configurations, characterizes their fibre structure, and links them to classical combinatorial structures, expanding the theory of coherent configurations.

Findings

P-polynomial coherent configurations can have at most two fibres.

Examples include distance-biregular graphs and strongly regular designs.

The paper establishes a connection between these configurations and classical combinatorial objects.

Abstract

We introduce the notion of P-polynomial coherent configurations and show that they can have at most two fibres. We then introduce a class of two-fibre coherent configurations which have two distinguished bases for the coherent algebra, similar to the Bose-Mesner algebra of an association scheme. Examples of these bipartite coherent configurations include the P-polynomial class of distance-biregular graphs, as well as quasi-symmetric designs and strongly regular designs.