Every 4-equivalenced association scheme is Frobenius

TL;DR

This paper proves that all 4-equivalenced association schemes are Frobenius, extending known results for smaller valencies and providing a new classification in algebraic combinatorics.

Contribution

It establishes that every 4-equivalenced association scheme is Frobenius, filling a gap in the classification of association schemes.

Findings

All 4-equivalenced association schemes are Frobenius.

Extends known results from k=2,3 to k=4.

Provides a new understanding of the structure of association schemes.

Abstract

For a positive integer $k$, we say that an association scheme $(\Omega,S)$ is $k$-equivalenced if each non-diagonal element of $S$ has valency $k$. $k$-equivalenced is weaker than pseudocyclic. It is known that every $k$-equivalenced association scheme is Frobenius when $k=2,3$ and every $4$-equivalenced association scheme is pseudocyclic. In this paper, we will show that every $4$-equivalenced association scheme is Frobenius.