Combinatorics Seminar

Given a graph G, let P_G denote the cone of positive semidefinite (psd) matrices with zero pattern according to G. Which powers preserve psd-ness when applied entrywise to all matrices in P_G?

In recent joint work with D. Guillot and B. Rajaratnam, we show how preserving positivity relates to the geometry of the graph G. This leads us to propose a novel graph invariant: the "critical exponent" of G. Our main result shows how this combinatorial invariant resolves the problem for all chordal graphs. We also report on progress for several families of non-chordal graphs.