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JCT
2016

Critical exponents of graphs

4 years 4 months ago
Critical exponents of graphs
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◦α := (aα ij) is positive semidefinite for every entrywise nonnegative n × n positive semidefinite matrix A = (aij) if and only if α is a positive integer or α ≥ 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 a...
Dominique Guillot, Apoorva Khare, Bala Rajaratnam
Added 06 Apr 2016
Updated 06 Apr 2016
Type Journal
Year 2016
Where JCT
Authors Dominique Guillot, Apoorva Khare, Bala Rajaratnam
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