{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:19:38Z","timestamp":1759335578106,"version":"3.41.2"},"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>Considering a connected graph $G$ with diameter $D$, we say that it is $k$-walk-regular, for a given integer $k$ $(0\\leq k \\leq D)$, if the number of walks of length $\\ell$ between any pair of vertices only depends on the distance between them, provided that this distance does not exceed $k$. Thus, for $k=0$, this definition coincides with that of walk-regular graph, where the number of cycles of length $\\ell$ rooted at a given vertex is a constant through all the graph. In the other extreme, for $k=D$, we get one of the possible definitions for a graph to be distance-regular. In this paper we show some algebraic characterizations of $k$-walk-regularity, which are based on the so-called local spectrum and predistance polynomials of $G$.<\/jats:p>","DOI":"10.37236\/136","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:32:22Z","timestamp":1578717142000},"source":"Crossref","is-referenced-by-count":14,"title":["On $k$-Walk-Regular Graphs"],"prefix":"10.37236","volume":"16","author":[{"given":"C.","family":"Dalf\u00f3","sequence":"first","affiliation":[]},{"given":"M. A.","family":"Fiol","sequence":"additional","affiliation":[]},{"given":"E.","family":"Garriga","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,4,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r47\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r47\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T03:04:53Z","timestamp":1579316693000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r47"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,4,22]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/136","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,4,22]]},"article-number":"R47"}}