Bounds on minimum semidefinite rank of graphs

Abstract

The minimum semidefinite rank of a graph is defined to be the minimum rank among all Hermitian positive semidefinite matrices associated to the graph. A problem of interest is to find upper and lower bounds for of a graph using known graph parameters such as the independence number and the minimum degree of the graph. We provide a sufficient condition for of a bipartite graph to equal its independence number. The delta conjecture gives an upper bound for of a graph in terms of its minimum degree. We present classes of graphs for which the delta conjecture holds.