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.