  The study of entrywise powers of matrices was originated by Loewner in the
pursuit of the Bieberbach conjecture. Since the work of FitzGerald and Horn
(1977), it is known that $A^{\circ \alpha} := (a_{ij}^\alpha)$ is positive
semidefinite for every entrywise nonnegative $n \times n$ positive semidefinite
matrix $A = (a_{ij})$ if and only if $\alpha$ is a positive integer or $\alpha
\geq n-2$. This surprising result naturally extends the Schur product theorem,
and demonstrates the existence of a sharp phase transition in preserving
positivity. In this paper, we study when entrywise powers preserve positivity
for matrices with structure of zeros encoded by graphs. To each graph is
associated an invariant called its "critical exponent", beyond which every
power preserves positivity. In our main result, we determine the critical
exponents of all chordal/decomposable graphs, and relate them to the geometry
of the underlying graphs. We then examine the critical exponent of important
families of non-chordal graphs such as cycles and bipartite graphs.
Surprisingly, large families of dense graphs have small critical exponents that
do not depend on the number of vertices of the graphs.
