{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:50Z","timestamp":1753893830143,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The $(n,k)$-arrangement graph $A(n,k)$ is a graph with all the $k$-permutations of an $n$-element set as vertices where two $k$-permutations are adjacent if they agree in exactly $k-1$ positions.\u00a0We introduce a cyclic decomposition for $k$-permutations and show that this gives rise to a very fine equitable partition\u00a0of $A(n,k)$. This equitable partition can be employed to compute the complete set of eigenvalues (of the adjacency matrix) of $A(n,k)$. Consequently, we determine the eigenvalues of $A(n,k)$ for small values of $k$.\u00a0Finally, we show that any eigenvalue of the Johnson graph $J(n,k)$ is an eigenvalue of $A(n,k)$ and that $-k$ is the smallest eigenvalue of $A(n,k)$ with multiplicity ${\\cal O}(n^k)$ for fixed $k$.<\/jats:p>","DOI":"10.37236\/3711","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:12:34Z","timestamp":1578705154000},"source":"Crossref","is-referenced-by-count":10,"title":["Cyclic Decomposition of $k$-Permutations and Eigenvalues of the Arrangement Graphs"],"prefix":"10.37236","volume":"20","author":[{"given":"Bai Fan","family":"Chen","sequence":"first","affiliation":[]},{"given":"Ebrahim","family":"Ghorbani","sequence":"additional","affiliation":[]},{"given":"Kok Bin","family":"Wong","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,11,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i4p22\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i4p22\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:10:54Z","timestamp":1579259454000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i4p22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,11,22]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2013,10,14]]}},"URL":"https:\/\/doi.org\/10.37236\/3711","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,11,22]]},"article-number":"P22"}}