{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T19:34:39Z","timestamp":1768678479928,"version":"3.49.0"},"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>A graph $G$ is said to be\u00a0determined by its generalized spectrum\u00a0(DGS for short) if\u00a0for any graph $H$, $H$ and $G$ are cospectral with cospectral\u00a0complements implies that $H$ is isomorphic to $G$. Wang and Xu (2006) gave some methods for determining\u00a0whether a family of graphs are DGS. In this paper, we shall review\u00a0some of the old results and present some new ones along this line\u00a0of research.More precisely, let $A$ be the adjacency matrix of a graph $G$,\u00a0and let $W=[e,Ae,\\cdots,A^{n-1}e]$ ($e$ is the all-one vector) be\u00a0its walk-matrix. Denote by $\\mathcal{G}_n$ the set of all\u00a0graphs on $n$ vertices with $\\det(W)\\neq 0$. We define a large\u00a0family of graphs $$\\mathcal{F}_n=\\{G\\in{\\mathcal{G}_n}|\\frac{\\det(W)}{2^{\\lfloorn\/2\\rfloor}}\\mbox{is square-free and }2^{\\lfloorn\/2\\rfloor+1}\\not|\\det(W)\\}$$ (which may have positive density\u00a0among all graphs, as suggested by some numerical experiments). The\u00a0main result of the paper shows that for any graph $G\\in\u00a0{\\mathcal{F}_n}$, if there is a rational orthogonal matrix $Q$\u00a0with $Qe=e$ such that $Q^TAQ$ is a (0,1)-matrix, then $2Q$ must be\u00a0an integral matrix (and hence, $Q$ has well-known structures). As\u00a0a consequence, we get the conclusion that almost all graphs in\u00a0$\\mathcal{F}_n$ are DGS.<\/jats:p>","DOI":"10.37236\/3748","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:13:16Z","timestamp":1578687196000},"source":"Crossref","is-referenced-by-count":23,"title":["Generalized Spectral Characterization of Graphs Revisited"],"prefix":"10.37236","volume":"20","author":[{"given":"Wei","family":"Wang","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,10,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i4p4\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i4p4\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T06:12:01Z","timestamp":1579241521000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i4p4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,10,21]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2013,10,14]]}},"URL":"https:\/\/doi.org\/10.37236\/3748","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,10,21]]},"article-number":"P4"}}