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A surprising result of FitzGerald and Horn (1977) shows that $A^{\circ\alpha} := (a_{ij}^\alpha)$ is positive semidefinite (p.s.d.) for everyentrywise 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 $G$, weconsider the refined problem of characterizing the set $\mathcal{H}_G$ ofentrywise powers preserving positivity for matrices with a zero pattern encodedby $G$. Using algebraic and combinatorial methods, we study how the geometry of$G$ influences the set $\mathcal{H}_G$. Our treatment provides new and excitingconnections between combinatorics and analysis, and leads us to introduce andcompute a new graph invariant called the critical exponent.

The critical exponent: a novel graph invariant