The Critical Exponent: A Novel Graph Invariant

by Apoorva Khare (Indian Institute of Science, Bangalore)

Tuesday 17 Oct 2017, 16:00 → 17:00 Asia/Kolkata

A-201 (STCS Seminar Room)

Given a graph $G$, let $\mathbb{P}_G$ denote the cone of positive semidefinite (psd) matrices, with non-negative entries, and zeros according to $G$. Which powers preserve psd-ness when applied entrywise to all matrices in $\mathbb{P}_G$? In recent work, joint 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.