# The Critical Exponent: A Novel Graph Invariant

Apoorva Khare

## Affiliation:

Indian Institute of Science
Department of Mathematics
Bangalore 560012

## Time:

Tuesday, 17 October 2017, 16:00 to 17:00

## Venue:

• A-201 (STCS Seminar Room)

## Organisers:

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.