Orthogonal representations of Steiner triple system incidence graphs

TL;DR

This paper investigates the minimum semidefinite rank of the Heawood graph, the incidence graph of the Fano plane, and explores extensions to larger Steiner triple systems, providing new insights into their orthogonal representations.

Contribution

It determines the minimum semidefinite rank of the Heawood graph and extends some techniques to larger Steiner triple system incidence graphs.

Findings

Minimum semidefinite rank of the Heawood graph is 10

Techniques extend to larger Steiner triple systems

Includes observations and open questions for general cases

Abstract

The Heawood graph is the point-block incidence graph of the Fano plane (the unique Steiner triple system of order 7). We show that the minimum semidefinite rank of this graph is 10. That is, 10 is the smallest number of complex dimensions in which this graph has a faithful orthogonal representation, i.e., an assignment of a vector to each vertex such that the edges occur between precisely those vertices given non-orthogonal pairs. Some of our techniques extend to the incidence graphs of Steiner triple systems of larger order, and we include some observations and questions about the more general case.