{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:29Z","timestamp":1753893869903,"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>Let $R_{n}$ denote the graph with vertex set consisting of the squares of an $n \\times n$ grid, with two squares of the grid adjacent when they lie in the same row or column. \u00a0This is the square rook's graph, and can also be thought of as the Cartesian product of two complete graphs of order $n$, or the line graph of the complete bipartite graph $K_{n,n}$. \u00a0In this paper we compute the Smith group and critical group of the graph $R_{n}$ and its complement. \u00a0This is equivalent to determining the Smith normal form of both the adjacency and Laplacian matrix of each of these graphs. \u00a0In doing so we verify a 1986 conjecture of Rushanan.<\/jats:p>","DOI":"10.37236\/5442","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:09:01Z","timestamp":1578686941000},"source":"Crossref","is-referenced-by-count":4,"title":["The Smith and Critical Groups of the Square Rook's Graph and its Complement"],"prefix":"10.37236","volume":"23","author":[{"given":"Joshua E.","family":"Ducey","sequence":"first","affiliation":[]},{"given":"Jonathan","family":"Gerhard","sequence":"additional","affiliation":[]},{"given":"Noah","family":"Watson","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,10,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i4p9\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i4p9\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:11:32Z","timestamp":1579237892000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i4p9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,10,14]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2016,10,14]]}},"URL":"https:\/\/doi.org\/10.37236\/5442","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2016,10,14]]},"article-number":"P4.9"}}