{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:08Z","timestamp":1753893848091,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>This paper describes the polytope $\\mathbf{P}_{k;N}$ of $i$-star counts, for all $i\\le k$, for graphs on $N$ nodes.\u00a0 The vertices correspond to graphs that are regular or as regular as possible.\u00a0 For even $N$ the polytope is a cyclic polytope, and for odd $N$ the polytope is well-approximated by a cyclic polytope.\u00a0 As $N$ goes to infinity, $\\mathbf{P}_{k;N}$ approaches the convex hull of the moment curve. The affine symmetry group of $\\mathbf{P}_{k;N}$ contains just a single non-trivial element, which corresponds to forming the complement of a graph.The results generalize to the polytope $\\mathbf{P}_{I;N}$ of $i$-star counts, for $i$ in some set $I$ of non-consecutive integers.\u00a0 In this case, $\\mathbf{P}_{I;N}$ can still be approximated by a cyclic polytope, but it is usually not a cyclic polytope itself.Polytopes of subgraph statistics characterize corresponding exponential random graph models.\u00a0 The elongated shape of the $k$-star polytope gives a qualitative explanation of some of the degeneracies found in such random graph models.<\/jats:p>","DOI":"10.37236\/4471","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T19:40:37Z","timestamp":1578685237000},"source":"Crossref","is-referenced-by-count":1,"title":["The Polytope of $k$-Star Densities"],"prefix":"10.37236","volume":"24","author":[{"given":"Johannes","family":"Rauh","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,1,20]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p4\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p4\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:06:38Z","timestamp":1579237598000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i1p4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1,20]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,1,20]]}},"URL":"https:\/\/doi.org\/10.37236\/4471","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,1,20]]},"article-number":"P1.4"}}