Parameters of Quotient-Polynomial Graphs

TL;DR

This paper characterizes quotient-polynomial graphs using specific parameters, enabling the generation of a database of such graphs with small valency and class, and identifies new feasible parameters for symmetric association schemes with particular eigenvalues.

Contribution

It introduces a parameter set for describing symmetric association schemes generated by their adjacency matrices, facilitating the construction of quotient-polynomial graphs and discovering new feasible schemes.

Findings

Generated a database of quotient-polynomial graphs with small valency and up to 6 classes.

Identified new feasible parameter sets for symmetric association schemes with noncyclotomic eigenvalues.

Provided a characterization linking parameters to the structure of quotient-polynomial graphs.

Abstract

Fiol has characterized quotient-polynomial graphs as precisely the connected graphs whose adjacency matrix generates the adjacency algebra of a symmetric association scheme. We show that a collection of non-negative integer parameters of size $d + \frac{d(d-1)}{2}$ is adequate for describing symmetric association schemes of class $d$ that are generated by the adjacency matrix of their first non-trivial relation. We use this to generate a database of the corresponding quotient-polynomial graphs that have small valency and up to 6 classes, and among these find new feasible parameter sets for symmetric association schemes with noncyclotomic eigenvalues.