On the minimum rank of a positive semidefinite matrix with a given graph

Matthew Booth, Philip Hackney, Benjamin Harris, Charles R. Johnson, Margaret Lay, Lon H. Mitchell, Sivaram K. Narayan, Amanda Pascoe, Kelly Steinmetz, Brian D. Sutton, and Wendy Wang

SIAM J. Matrix Anal. Appl. 30 (2008), no. 2, 731–740.

Let P(G) be the set of all positive semidefinite matrices whose graph is G, and msr(G) be the minimum rank of all matrices in P(G). Upper and lower bounds for msr(G) are given and used to determine msr(G) for some well known graphs, including chordal graphs, and for all simple graphs on less than seven vertices.