[{"abstract":[{"lang":"eng","text":"Given a bipartite graph G, the graphical matrix space SG consists of\r\nmatrices whose non-zero entries can only be at those positions corresponding to edges in G. Tutte (J. London Math. Soc., 1947), Edmonds\r\n(J. Res. Nat. Bur. Standards Sect. B, 1967) and Lov´asz (FCT, 1979) observed connections between perfect matchings in G and full-rank matrices\r\nin SG. Dieudonn´e (Arch. Math., 1948) proved a tight upper bound on\r\nthe dimensions of those matrix spaces containing only singular matrices.\r\nThe starting point of this paper is a simultaneous generalization of these\r\ntwo classical results: we show that the largest dimension over subspaces\r\nof SG containing only singular matrices is equal to the maximum size over\r\nsubgraphs of G without perfect matchings, based on Meshulam’s proof of\r\nDieudonn´e’s result (Quart. J. Math., 1985).\r\nStarting from this result, we go on to establish more connections\r\nbetween properties of graphs and matrix spaces. For example, we\r\nestablish connections between acyclicity and nilpotency, between strong\r\nconnectivity and irreducibility, and between isomorphism and\r\nconjugacy/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\r\n(for induced subgraphs and restrictions). Some correspondences lead to\r\nintriguing generalizations of classical results, such as Dieudonn´e’s result\r\nmentioned above, and a celebrated theorem of Gerstenhaber regarding the\r\nlargest dimension of nil matrix spaces (Amer. J. Math., 1958).\r\nFinally, we show some implications of our results to quantum information and present open problems in computational complexity motivated\r\nby these results."}],"oa":1,"language":[{"iso":"eng"}],"OA_type":"green","scopus_import":"1","_id":"22162","issue":"2","date_updated":"2026-07-08T10:44:50Z","citation":{"ista":"Li Y, Qiao Y, Wigderson A, Wigderson Y, Zhang C. 2023. Connections between graphs and matrix spaces. Israel Journal of Mathematics. 256(2), 513–580.","ama":"Li Y, Qiao Y, Wigderson A, Wigderson Y, Zhang C. Connections between graphs and matrix spaces. <i>Israel Journal of Mathematics</i>. 2023;256(2):513-580. doi:<a href=\"https://doi.org/10.1007/s11856-023-2515-7\">10.1007/s11856-023-2515-7</a>","apa":"Li, Y., Qiao, Y., Wigderson, A., Wigderson, Y., &#38; Zhang, C. (2023). Connections between graphs and matrix spaces. <i>Israel Journal of Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s11856-023-2515-7\">https://doi.org/10.1007/s11856-023-2515-7</a>","ieee":"Y. Li, Y. Qiao, A. Wigderson, Y. Wigderson, and C. Zhang, “Connections between graphs and matrix spaces,” <i>Israel Journal of Mathematics</i>, vol. 256, no. 2. Springer Nature, pp. 513–580, 2023.","mla":"Li, Yinan, et al. “Connections between Graphs and Matrix Spaces.” <i>Israel Journal of Mathematics</i>, vol. 256, no. 2, Springer Nature, 2023, pp. 513–80, doi:<a href=\"https://doi.org/10.1007/s11856-023-2515-7\">10.1007/s11856-023-2515-7</a>.","short":"Y. Li, Y. Qiao, A. Wigderson, Y. Wigderson, C. Zhang, Israel Journal of Mathematics 256 (2023) 513–580.","chicago":"Li, Yinan, Youming Qiao, Avi Wigderson, Yuval Wigderson, and Chuanqi Zhang. “Connections between Graphs and Matrix Spaces.” <i>Israel Journal of Mathematics</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s11856-023-2515-7\">https://doi.org/10.1007/s11856-023-2515-7</a>."},"quality_controlled":"1","extern":"1","doi":"10.1007/s11856-023-2515-7","date_created":"2026-06-29T10:52:37Z","publication_status":"published","type":"journal_article","day":"01","article_type":"original","publication":"Israel Journal of Mathematics","OA_place":"repository","publisher":"Springer Nature","arxiv":1,"status":"public","intvolume":"       256","month":"09","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2206.04815","open_access":"1"}],"publication_identifier":{"eissn":["1565-8511"],"issn":["0021-2172"]},"external_id":{"arxiv":["2206.04815"]},"author":[{"full_name":"Li, Yinan","first_name":"Yinan","last_name":"Li"},{"first_name":"Youming","full_name":"Qiao, Youming","last_name":"Qiao"},{"first_name":"Avi","full_name":"Wigderson, Avi","last_name":"Wigderson"},{"full_name":"Wigderson, Yuval","first_name":"Yuval","last_name":"Wigderson","id":"2d0023a0-1567-11f0-833d-d5c1e476d4b5"},{"last_name":"Zhang","full_name":"Zhang, Chuanqi","first_name":"Chuanqi"}],"title":"Connections between graphs and matrix spaces","article_processing_charge":"No","volume":256,"oa_version":"Preprint","date_published":"2023-09-01T00:00:00Z","year":"2023","page":"513-580","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"}]