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Connections between graphs and matrix spaces

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Abstract

Given a bipartite graph G, the graphical matrix space \(\cal{S}_{G}\) consists of matrices whose non-zero entries can only be at those positions corresponding to edges in G. Tutte (J. London Math. Soc., 1947), Edmonds (J. Res. Nat. Bur. Standards Sect. B, 1967) and Lovász (FCT, 1979) observed connections between perfect matchings in G and full-rank matrices in \(\cal{S}_{G}\). Dieudonné (Arch. Math., 1948) proved a tight upper bound on the dimensions of those matrix spaces containing only singular matrices. The starting point of this paper is a simultaneous generalization of these two classical results: we show that the largest dimension over subspaces of \(\cal{S}_{G}\) containing only singular matrices is equal to the maximum size over subgraphs of G without perfect matchings, based on Meshulam’s proof of Dieudonné’s result (Quart. J. Math., 1985).

Starting from this result, we go on to establish more connections between properties of graphs and matrix spaces. For example, we establish connections between acyclicity and nilpotency, between strong connectivity and irreducibility, and between isomorphism and conjugacy/congruence. For each connection, we study three types of correspondences, namely the basic correspondence, the inherited correspondence (for subgraphs and subspaces), and the induced correspondence (for induced subgraphs and restrictions). Some correspondences lead to intriguing generalizations of classical results, such as Dieudonné’s result mentioned above, and a celebrated theorem of Gerstenhaber regarding the largest dimension of nil matrix spaces (Amer. J. Math., 1958).

Finally, we show some implications of our results to quantum information and present open problems in computational complexity motivated by these results.

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Acknowledgments

Y. Q. would like to thank George Glauberman for helping him with Proposition 7.1. We would also like to thank an anonymous referee for helpful comments and intriguing suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics and Hubei Computational Science Key Laboratory, Wuhan University, Wuhan, 430072, China

    Yinan Li

  2. Centre for Quantum Software and Information, University of Technology Sydney, Sydney, NSW, 2007, Australia

    Youming Qiao & Chuanqi Zhang

  3. School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540, USA

    Avi Wigderson

  4. Department of Mathematics, Tel Aviv University, Tel Aviv, 6997801, Israel

    Yuval Wigderson

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  1. Yinan Li
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  2. Youming Qiao
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  4. Yuval Wigderson
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Correspondence to Yuval Wigderson.

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Dedicated to Nati Linial on the occasion of his 70th birthday

Research supported in part by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Number JPMXS0120319794.

Research supported in part by Australian Research Council DP200100950.

Research supported in part by NSF grant CCF-1900460.

Research supported in part by NSF GRFP Grant DGE-1656518, ERC Consolidator Grant 863438, ERC Starting Grant 101044123 and NSF-BSF Grant 20196.

Research supported by Australian Research Council DP200100950 and the Sydney Quantum Academy, Sydney, NSW, Australia.

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Li, Y., Qiao, Y., Wigderson, A. et al. Connections between graphs and matrix spaces. Isr. J. Math. 256, 513–580 (2023). https://doi.org/10.1007/s11856-023-2515-7

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