We study graph-theoretic properties of random 0/1 polytopes. Specifically, let \ud835\udc44\ud835\udc5b\ud835\udc5d ⊆{0,1}\ud835\udc5b be a random subset where each point is included independently with probability \ud835\udc5d, and consider the graph \ud835\udc3a\ud835\udc5d of the polytope conv\u2061(\ud835\udc44\ud835\udc5b\ud835\udc5d). We provide a short and combinatorial proof that \ud835\udc5d =2−\ud835\udc5b/2 is a threshold for when the edge density of \ud835\udc3a\ud835\udc5d is 1, a result originally due to Kaibel and Remshagen. We next resolve an open question from their paper by showing that for \ud835\udc5d \u2a7d2−\ud835\udc5b/2−\ud835\udc5c\u2061(1), \ud835\udc3a\ud835\udc5d exhibits strong edge expansion. In particular, we prove that, with high probability, every vertex has degree (1−\ud835\udc5c\u2061(1))|\u2062\ud835\udc44\ud835\udc5b\ud835\udc5d|. Lastly, we determine the threshold for \ud835\udc3a\ud835\udc5d being a clique, strengthening a result of Bondarenko and Brodskiy. We show that with high probability, if \ud835\udc5d \u2a7e2−\ud835\udeff\u2062\ud835\udc5b+\ud835\udc5c\u2061(1), then \ud835\udc3a\ud835\udc5d is not a clique, and if \ud835\udc5d \u2a7d2−\ud835\udeff\u2062\ud835\udc5b−\ud835\udc5c\u2061(1), then \ud835\udc3a\ud835\udc5d is a clique, where \ud835\udeff ≈0.8295. Our approach combines a combinatorial characterization of edges in graphs arising from polytopes with the Kim–Vu polynomial concentration inequality.
", "doi": "10.1112/blms.70456", "issn": "0024-6093", "publisher": "Wiley", "publication": "Bulletin of the London Mathematical Society", "publication_date": "2026-08", "series_number": "8", "volume": "58", "issue": "8", "pages": "e70456" }, { "id": "authors:gxg43-khm58", "collection": "authors", "collection_id": "gxg43-khm58", "cite_using_url": "https://authors.library.caltech.edu/records/gxg43-khm58", "type": "article", "title": "Spectrahedral Geometry of Graph Sparsifiers", "author": [ { "family_name": "Babecki", "given_name": "Catherine", "orcid": "0000-0003-4107-0737", "clpid": "Babecki-Catherine" }, { "family_name": "Steinerberger", "given_name": "Stefan", "orcid": "0000-0002-7745-4217", "clpid": "Steinerberger-Stefan" }, { "family_name": "Thomas", "given_name": "Rekha R.", "orcid": "0000-0003-2189-2303", "clpid": "Thomas-Rekha-R" } ], "abstract": "We propose an approach to graph sparsification based on the idea of preserving the smallest k eigenvalues and eigenvectors of the Graph Laplacian. This is motivated by the fact that small eigenvalues and their associated eigenvectors tend to be more informative of the global structure and geometry of the graph than larger eigenvalues and their eigenvectors. The set of all weighted subgraphs of a graph G that have the same first k eigenvalues (and eigenvectors) as G is the intersection of a polyhedron with a cone of positive semidefinite matrices. We discuss the geometry of these sets and deduce the natural scale of k. Various families of graphs illustrate our construction.
", "doi": "10.1137/23M1610069", "issn": "0895-4801", "publisher": "Society for Industrial & Applied Mathematics (SIAM)", "publication": "SIAM Journal on Discrete Mathematics", "publication_date": "2025-02-18", "series_number": "1", "volume": "39", "issue": "1", "pages": "449-483" }, { "id": "authors:mx1n4-eyw65", "collection": "authors", "collection_id": "mx1n4-eyw65", "cite_using_url": "https://authors.library.caltech.edu/records/mx1n4-eyw65", "type": "article", "title": "Sparse graphical designs via linear programming", "author": [ { "family_name": "Al-Thani", "given_name": "Hessa", "orcid": "0000-0002-1777-1069" }, { "family_name": "Babecki", "given_name": "Catherine", "orcid": "0000-0003-4107-0737", "clpid": "Babecki-Catherine" }, { "family_name": "Mart\u00ednez Mori", "given_name": "J. Carlos" } ], "abstract": "We propose an approach to graph sparsification based on the idea of preserving the smallest k eigenvalues and eigenvectors of the Graph Laplacian. This is motivated by the fact that small eigenvalues and their associated eigenvectors tend to be more informative of the global structure and geometry of the graph than larger eigenvalues and their eigenvectors. The set of all weighted subgraphs of a graph G that have the same first k eigenvalues (and eigenvectors) as G is the intersection of a polyhedron with a cone of positive semidefinite matrices. We discuss the geometry of these sets and deduce the natural scale of k. Various families of graphs illustrate our construction.
", "doi": "10.48550/arXiv.2306.06204", "publication_date": "2023-06-09" } ]