Extremal Positive Semidefinite Matrices for Graphs without K_5 Minors

Liam Solus, Caroline Uhler, Ruriko Yoshida

Abstract: For a graph G with p vertices the closed convex cone 𝕊p⪰0(G) consists of all real positive semidefinite p×p matrices with zeros in the off-diagonal entries corresponding to nonedges of G. The extremal rays of this cone and their associated ranks have applications to matrix completion problems, maximum likelihood estimation in Gaussian graphical models in statistics, and Gauss elimination for sparse matrices. For a graph G without K5 minors, we show that the normal vectors to the facets of the (±1)-cut polytope of G specify the off-diagonal entries of extremal matrices in 𝕊p⪰0(G). We also prove that the constant term of the linear equation of each facet-supporting hyperplane is the rank of its corresponding extremal matrix in 𝕊p⪰0(G). Furthermore, we show that if G is series-parallel then this gives a complete characterization of all possible extremal ranks of 𝕊p⪰0(G), consequently solving the sparsity order problem for series-parallel graphs.