{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T17:45:49Z","timestamp":1772214349226,"version":"3.50.1"},"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>Let $X$ be a graph on $n$ vertices with adjacency matrix $A$ and let $H(t)$ denote the matrix-valued function $\\exp(iAt)$.  If $u$ and $v$ are distinct vertices in $X$, we say perfect state transfer from $u$ to $v$ occurs if there is a time $\\tau$ such that $|H(\\tau)_{u,v}|=1$. If $u\\in V(X)$ and there is a time $\\sigma$ such that $|H(\\sigma)_{u,u}|=1$, we say $X$ is periodic at $u$ with period $\\sigma$. It is not difficult to show that if the ratio of distinct non-zero eigenvalues of $X$ is always rational, then $X$ is periodic. We show that the converse holds, from which it follows that a regular graph is periodic if and only if its eigenvalues are distinct.  For a class of graphs $X$ including all vertex-transitive graphs we prove that, if perfect state transfer occurs at time $\\tau$, then $H(\\tau)$ is a scalar multiple of a permutation matrix of order two with no fixed points.  Using certain Hadamard matrices, we construct a new infinite family of graphs on which perfect state transfer occurs.<\/jats:p>","DOI":"10.37236\/510","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:56:50Z","timestamp":1578715010000},"source":"Crossref","is-referenced-by-count":52,"title":["Periodic Graphs"],"prefix":"10.37236","volume":"18","author":[{"given":"Chris","family":"Godsil","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,1,26]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p23\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p23\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:17:38Z","timestamp":1579303058000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p23"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1,26]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/510","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1,26]]},"article-number":"P23"}}