# Partially metric association schemes with a multiplicity three

## TL;DR

This paper classifies symmetric partially metric association schemes with multiplicity three, linking them to well-known graphs and constructing new families of graphs with specific eigenvalue properties.

## Contribution

It provides a complete classification of symmetric partially metric association schemes with multiplicity three and constructs new infinite families of graphs with specific spectral characteristics.

## Key findings

- Classified symmetric partially metric association schemes with multiplicity three.
- Connected to well-known graphs like Platonic solids and certain 2-arc-transitive covers.
- Constructed infinite families of cubic arc-transitive 2-walk-regular graphs with eigenvalue multiplicity three.

## Abstract

An association scheme is called partially metric if it has a connected relation whose distance-two relation is also a relation of the scheme. In this paper we determine the symmetric partially metric association schemes with a multiplicity three. Besides the association schemes related to regular complete $4$-partite graphs, we obtain the association schemes related to the Platonic solids, the bipartite double scheme of the dodecahedron, and three association schemes that are related to well-known $2$-arc-transitive covers of the cube: the M\"{o}bius-Kantor graph, the Nauru graph, and the Foster graph F048A. In order to obtain this result, we also determine the symmetric association schemes with a multiplicity three and a connected relation with valency three. Moreover, we construct an infinite family of cubic arc-transitive $2$-walk-regular graphs with an eigenvalue with multiplicity three that give rise to non-commutative association schemes with a symmetric relation of valency three and an eigenvalue with multiplicity three.