{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:22Z","timestamp":1753893802604,"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 simple connected graph and $\\mu_1(G) \\geq \\mu_2(G) \\geq \\cdots \\geq \\mu_n(G)$ be the Laplacian eigenvalues of $G$. Let $\\overline{G}$ be the complement of $G$. Einollahzadeh et al.[J. Combin. Theory Ser. B, 151(2021), 235\u2013249] proved that $\\mu_{n-1}(G)+\\mu_{n-1}(\\overline{G})\\geq 1$. Grij\u00f2 et al. [Discrete Appl. Math., 267(2019), 176\u2013183] conjectured that $\\mu_{n-2}(G)+\\mu_{n-2}(\\overline{G})\\geq 2$ for any graph and proved it to be true for some graphs. In this paper, we prove $\\mu_{n-2}(G)+\\mu_{n-2}(\\overline{G})\\geq 2$ is true for some new graphs. Furthermore, we propose a more general conjecture that $\\mu_k(G)+\\mu_k(\\overline{G})\\geq n-k$ holds for any graph $G$, with equality if and only if $G$ or $\\overline{G}$ is isomorphic to $K_{n-k}\\vee H$, where $H$ is a disconnected graph on $k$ vertices and has at least $n-k+1$ connected components. And we prove that it is true for $k\\leq \\frac{n+1}{2}$, for unicyclic graphs, bicyclic graphs, threshold graphs, bipartite graphs, regular graphs, complete multipartite graphs and c-cyclic graphs when $n\\geq 2c+8$.<\/jats:p>","DOI":"10.37236\/12008","type":"journal-article","created":{"date-parts":[[2024,2,9]],"date-time":"2024-02-09T15:45:34Z","timestamp":1707493534000},"source":"Crossref","is-referenced-by-count":0,"title":["Nordhaus-Gaddum Type Inequalities for the $k$th Largest Laplacian Eigenvalues"],"prefix":"10.37236","volume":"31","author":[{"given":"Wen-Jun","family":"Li","sequence":"first","affiliation":[]},{"given":"Ji-Ming","family":"Guo","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2024,2,9]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i1p31\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i1p31\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,9]],"date-time":"2024-02-09T15:45:35Z","timestamp":1707493535000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v31i1p31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,9]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1,12]]}},"URL":"https:\/\/doi.org\/10.37236\/12008","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2024,2,9]]},"article-number":"P1.31"}}