  A surprising result of FitzGerald and Horn (1977) shows that $A^{\circ
\alpha} := (a_{ij}^\alpha)$ is positive semidefinite (p.s.d.) for every
entrywise nonnegative $n \times n$ p.s.d. matrix $A = (a_{ij})$ if and only if
$\alpha$ is a positive integer or $\alpha \geq n-2$. Given a graph $G$, we
consider the refined problem of characterizing the set $\mathcal{H}_G$ 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 $\mathcal{H}_G$. 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.
