Orthogonal representations of Steiner triple system incidence graphs

Abstract

The unique Steiner triple system of order 7 has a point-block incidence graph known as the Heawood graph. Motivated by questions in combinatorial matrix theory, we consider the problem of constructing a faithful orthogonal representation of this graph, i.e., an assignment of a vector in Cd to each vertex such that two vertices are adjacent precisely when assigned nonorthogonal vectors. We show that d=10 is the smallest number of dimensions in which such a representation exists, a value known as the minimum semidefinite rank of the graph, and give such a representation in 10 real dimensions. We then show how the same approach gives a lower bound on this parameter for the incidence graph of any Steiner triple system, and highlight some questions concerning the general upper bound.