Critical exponents of graphs

TL;DR

This paper investigates the preservation of positive semidefiniteness under entrywise powers of matrices with zero patterns defined by graphs, identifying critical exponents for various graph classes.

Contribution

It determines the critical exponents for all chordal graphs and relates them to graph geometry, extending understanding of positivity preservation.

Findings

Critical exponents are determined for all chordal graphs.

Large dense graphs can have small, vertex-independent critical exponents.

The study reveals phase transitions in positivity preservation related to graph structure.

Abstract

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.