Distance-Regular Graphs with an Eigenvalue of Multiplicity Four

Let G be a distance-regular graph. If G has an eigenvalue θ of multiplicity m (≥ 2), then G has a natural representation in Rm. By studying the geometric properties of the image configuration in Rm, we can obtain considerable information about the graph-theoretic properties of G itself. This sets the basis for classifying distance-regular graphs by their eigenvalue multiplicities. It is known that the distance-regular graphs with an eigenvalue of multiplicity three are exactly the five Platonic solids plus all complete 4-partite regular graphs. In this paper we classify the distance-regular graphs with an eigenvalue of multiplicity four.