{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:14Z","timestamp":1753893854309,"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>A normally regular digraph with parameters $(v,k,\\lambda,\\mu)$ is a\u00a0directed graph on $v$ vertices whose adjacency matrix\u00a0$A$ satisfies the equation $AA^t=k I+\\lambda (A+A^t)+\\mu(J-I-A-A^t)$. This means that every vertex has out-degree $k$, a pair\u00a0of non-adjacent vertices have $\\mu$ common out-neighbours, a pair of\u00a0vertices connected by an edge in one direction have $\\lambda$ common\u00a0out-neighbours and a pair of vertices connected by edges in both\u00a0directions have $2\\lambda-\\mu$ common out-neighbours. We often assume\u00a0that two vertices can not be connected in both directions. We prove that the adjacency matrix of a normally regular digraph is\u00a0normal. A connected $k$-regular digraph with normal adjacency matrix is a normally regular digraph if and only if all eigenvalues other than $k$ are on one circle in the complex plane.\u00a0We prove several non-existence results, structural characterizations, and constructions of normally regular digraphs. In many cases these graphs\u00a0are Cayley graphs of abelian groups and the construction is then based\u00a0on a generalization of difference sets.We also show connections to other combinatorial objects:\u00a0strongly regular graphs, symmetric 2-designs and association schemes.<\/jats:p>","DOI":"10.37236\/4798","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T17:53:54Z","timestamp":1578678834000},"source":"Crossref","is-referenced-by-count":5,"title":["Normally Regular Digraphs"],"prefix":"10.37236","volume":"22","author":[{"given":"Leif K","family":"J\u00f8rgensen","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2015,10,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i4p21\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i4p21\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:08:46Z","timestamp":1579237726000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i4p21"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,10,30]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2015,10,16]]}},"URL":"https:\/\/doi.org\/10.37236\/4798","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2015,10,30]]},"article-number":"P4.21"}}