The minimum semidefinite rank of a triangle-free graph

Abstract

AbstractWe employ a result of Moshe Rosenfeld to show that the minimum semidefinite rank of a triangle-free graph with no isolated vertex must be at least half the number of its vertices. We define a Rosenfeld graph to be such a graph that achieves equality in this bound, and we explore the structure of these special graphs. Their structure turns out to be intimately connected with the zero–nonzero patterns of the unitary matrices. Finally, we suggest an exploration of the connection between the girth of a graph and its minimum semidefinite rank, and provide a conjecture in this direction