{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:07:29Z","timestamp":1758823649061,"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>A graph $G$ is well-covered if every maximal independent set has the same cardinality.  Let $s_k$ denote the number of independent sets of cardinality $k$, and define the independence polynomial of $G$ to be  $S(G,z) = \\sum s_kz^k$.  This paper develops a new graph theoretic operation called power magnification that preserves well-coveredness and has the effect of multiplying an independence polynomial by $z^c$ where $c$ is a positive integer.  We will apply power magnification to the recent Roller-Coaster Conjecture of Michael and Traves, proving in our main theorem that for sufficiently large independence number $\\alpha$, it is possible to find well-covered graphs with the last $(.17)\\alpha$ terms of the independence sequence in any given linear order.  Also, we will give a simple proof of a result due to Alavi, Malde, Schwenk, and Erd\u0151s on possible linear orderings of the independence sequence of not-necessarily well-covered graphs, and we will prove the Roller-Coaster Conjecture in full for independence number $\\alpha \\leq 11$.  Finally, we will develop two new graph operations that preserve well-coveredness and have interesting effects on the independence polynomial. <\/jats:p>","DOI":"10.37236\/1798","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:21:05Z","timestamp":1578702065000},"source":"Crossref","is-referenced-by-count":7,"title":["Operations on Well-Covered Graphs and the Roller-Coaster Conjecture"],"prefix":"10.37236","volume":"11","author":[{"given":"Philip","family":"Matchett","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2004,7,1]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r45\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r45\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T00:02:06Z","timestamp":1579305726000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i1r45"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,7,1]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2004,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/1798","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2004,7,1]]},"article-number":"R45"}}