# Extremal positive semidefinite matrices whose sparsity pattern is given by graphs without K5 minors

Solus LT, Uhler C, Yoshida R. 2016. Extremal positive semidefinite matrices whose sparsity pattern is given by graphs without K5 minors. Linear Algebra and Its Applications. 509, 247–275.

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Abstract

For a graph G with p vertices the closed convex cone S⪰0(G) consists of all real positive semidefinite p×p matrices whose sparsity pattern is given by G, that is, those matrices with zeros in the off-diagonal entries corresponding to nonedges of G. The extremal rays of this cone and their associated ranks have applications to matrix completion problems, maximum likelihood estimation in Gaussian graphical models in statistics, and Gauss elimination for sparse matrices. While the maximum rank of an extremal ray in S⪰0(G), known as the sparsity order of G, has been characterized for different classes of graphs, we here study all possible extremal ranks of S⪰0(G). We investigate when the geometry of the (±1)-cut polytope of G yields a polyhedral characterization of the set of extremal ranks of S⪰0(G). For a graph G without K5 minors, we show that appropriately chosen normal vectors to the facets of the (±1)-cut polytope of G specify the off-diagonal entries of extremal matrices in S⪰0(G). We also prove that for appropriately chosen scalars the constant term of the linear equation of each facet-supporting hyperplane is the rank of its corresponding extremal matrix in S⪰0(G). Furthermore, we show that if G is series-parallel then this gives a complete characterization of all possible extremal ranks of S⪰0(G). Consequently, the sparsity order problem for series-parallel graphs can be solved in terms of polyhedral geometry.

Publishing Year

Date Published

2016-11-15

Journal Title

Linear Algebra and Its Applications

Acknowledgement

We wish to thank Alexander Engström and Bernd Sturmfels for various valuable discussions and insights. We also thank the two anonymous referees for their thoughtful feedback on the paper. CU was partially supported by the Austrian Science Fund (FWF) Y 903-N35.

Volume

509

Page

247 - 275

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### Cite this

Solus LT, Uhler C, Yoshida R. Extremal positive semidefinite matrices whose sparsity pattern is given by graphs without K5 minors.

*Linear Algebra and Its Applications*. 2016;509:247-275. doi:10.1016/j.laa.2016.07.026Solus, L. T., Uhler, C., & Yoshida, R. (2016). Extremal positive semidefinite matrices whose sparsity pattern is given by graphs without K5 minors.

*Linear Algebra and Its Applications*. Elsevier. https://doi.org/10.1016/j.laa.2016.07.026Solus, Liam T, Caroline Uhler, and Ruriko Yoshida. “Extremal Positive Semidefinite Matrices Whose Sparsity Pattern Is given by Graphs without K5 Minors.”

*Linear Algebra and Its Applications*. Elsevier, 2016. https://doi.org/10.1016/j.laa.2016.07.026.L. T. Solus, C. Uhler, and R. Yoshida, “Extremal positive semidefinite matrices whose sparsity pattern is given by graphs without K5 minors,”

*Linear Algebra and Its Applications*, vol. 509. Elsevier, pp. 247–275, 2016.Solus, Liam T., et al. “Extremal Positive Semidefinite Matrices Whose Sparsity Pattern Is given by Graphs without K5 Minors.”

*Linear Algebra and Its Applications*, vol. 509, Elsevier, 2016, pp. 247–75, doi:10.1016/j.laa.2016.07.026.**All files available under the following license(s):**

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