{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T16:19:03Z","timestamp":1759940343659,"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>For a nilpotent group $G$, let $\\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\\Xi(G)$ has vertex set $G \\setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\\Xi^+(G)$ be the subgraph of $\\Xi(G)$ induced by its non-isolated vertices. We show that if $\\Xi(G)$ has an edge, then $\\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\\Xi(G) = \\Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\\Xi(G)$ in more detail.<\/jats:p>","DOI":"10.37236\/9802","type":"journal-article","created":{"date-parts":[[2021,2,11]],"date-time":"2021-02-11T10:11:51Z","timestamp":1613038311000},"source":"Crossref","is-referenced-by-count":4,"title":["The Non-Commuting, Non-Generating Graph of a Nilpotent Group"],"prefix":"10.37236","volume":"28","author":[{"given":"Peter","family":"Cameron","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Saul","family":"Freedman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Colva","family":"Roney-Dougal","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2021,1,29]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i1p16\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i1p16\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,11]],"date-time":"2021-02-11T10:11:51Z","timestamp":1613038311000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v28i1p16"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,29]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1,14]]}},"URL":"https:\/\/doi.org\/10.37236\/9802","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2021,1,29]]},"article-number":"P1.16"}}