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Séminaire Lotharingien de Combinatoire, 78B.62 (2017), 12 pp.

# Dominique Guillot, Apoorva
Khare and Bala Rajaratnam

# The Critical Exponent: a Novel Graph Invariant

**Abstract.**
A surprising result of FitzGerald and Horn (1977) shows that
*A*^{o
α} := (*a*_{ij}^{α})
is positive semidefinite (p.s.d.) for
every entrywise nonnegative *n* x *n* p.s.d. matrix
*A* = (*a*_{ij})
if and only if α is a positive integer or
α >=
*n*-2. Given a graph *G*, we consider the refined problem of
characterizing the set **H**_{G}
of entrywise powers preserving
positivity for matrices with a zero pattern encoded by *G*. Using
algebraic and combinatorial methods, we study how the geometry of *G*
influences the set **H**_{G}. Our treatment provides new and
exciting connections between combinatorics and analysis, and leads us
to introduce and compute a new graph invariant called the
*critical exponent*.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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