Parameters of Quotient-Polynomial Graphs

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.