{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:00Z","timestamp":1753893840866,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The enumeration of independent sets in graphs with various restrictions has been a topic of much interest of late.\u00a0 Let $i(G)$ be the number of independent sets in a graph $G$ and let $i_t(G)$ be the number of independent sets in $G$ of size $t$.\u00a0 Kahn used entropy to show that if $G$ is an $r$-regular bipartite graph with $n$ vertices, then $i(G)\\leq i(K_{r,r})^{n\/2r}$.\u00a0 Zhao used bipartite double covers to extend this bound to general $r$-regular graphs.\u00a0 Galvin proved that if $G$ is a graph with $\\delta(G)\\geq \\delta$ and $n$ large enough, then $i(G)\\leq i(K_{\\delta,n-\\delta})$.\u00a0 In this paper, we prove that if $G$ is a bipartite graph on $n$ vertices with $\\delta(G)\\geq\\delta$ where $n\\geq 2\\delta$, then $i_t(G)\\leq i_t(K_{\\delta,n-\\delta})$ when $t\\geq 3$.\u00a0 We note that this result cannot be extended to $t=2$ (and is trivial for $t=0,1$).\u00a0 Also, we use Kahn's entropy argument and Zhao's extension to prove that if $G$ is a graph with $n$ vertices, $\\delta(G)\\geq\\delta$, and $\\Delta(G)\\leq \\Delta$, then $i(G)\\leq i(K_{\\delta,\\Delta})^{n\/2\\delta}$.<\/jats:p>","DOI":"10.37236\/2722","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:23:46Z","timestamp":1578713026000},"source":"Crossref","is-referenced-by-count":13,"title":["Independent Sets in Graphs with Given Minimum Degree"],"prefix":"10.37236","volume":"19","author":[{"given":"James","family":"Alexander","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jonathan","family":"Cutler","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tim","family":"Mink","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2012,9,27]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i3p37\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i3p37\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T22:30:43Z","timestamp":1579300243000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i3p37"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,9,27]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2012,7,12]]}},"URL":"https:\/\/doi.org\/10.37236\/2722","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2012,9,27]]},"article-number":"P37"}}