{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:24Z","timestamp":1753893864964,"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>Let $G$ be a graph. It is well known that the maximum multiplicity of a root of the matching polynomial $\\mu(G,x)$ is at most the minimum number of vertex disjoint paths needed to cover the vertex set of $G$. Recently, a necessary and sufficient condition for which this bound is tight was found for trees. In this paper, a similar structural characterization is proved for any graph. To accomplish this, we extend the notion of a $(\\theta,G)$-extremal path cover (where $\\theta$ is a root of $\\mu(G,x)$) which was first introduced for trees to general graphs. Our proof makes use of the analogue of the Gallai-Edmonds Structure Theorem for general root. By way of contrast, we also show that the difference between the minimum size of a path cover and the maximum multiplicity of matching polynomial roots can be arbitrarily large.<\/jats:p>","DOI":"10.37236\/525","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:53:28Z","timestamp":1578714808000},"source":"Crossref","is-referenced-by-count":3,"title":["Maximum Multiplicity of Matching Polynomial Roots and Minimum Path Cover in General Graphs"],"prefix":"10.37236","volume":"18","author":[{"given":"Cheng Yeaw","family":"Ku","sequence":"first","affiliation":[]},{"given":"Kok Bin","family":"Wong","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2011,2,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p38\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p38\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:17:02Z","timestamp":1579303022000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p38"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,2,14]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/525","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,2,14]]},"article-number":"P38"}}