Relation Algebra Representations from Distance-Regular Graphs

We describe a general method for constructing representations of finite integral symmetric relation algebras from distance-regular graphs. Given a distance-regular graph of diameter $d$, the distances between vertices induces a coloring of the complete graph with $d$ colors, and we show that this coloring yields a representation of finite integral symmetric relation algebra on $d+1$ atoms. We then introduce a necessary and sufficient condition for when such a representation is algebraic, proving that this occurs if and only if the distance-regular graph is also distance-transitive. We study the diameter-3 case of this method in detail, and we express a condition for the representation's mandatory cycles in terms of the distance-regular graph's intersection array. We apply this result to give a positive answer to an open question of Roger Maddux; namely, whether the relation algebra $30_{65}$ has a representation on a finite set. The representation is given on 42 points, and arises from the second subconstituent of the Hoffman-Singleton graph. We further use this method to describe an infinite class of finite representations of $26_{65}$ and the smallest possible representation of $31_{65}$.