{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T17:32:19Z","timestamp":1762191139761,"version":"build-2065373602"},"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>For a graph $G$ of $n$ vertices, let $\\mu_{1}(G)$ be its largest Laplacian eigenvalue. It was conjectured by Ashraf et al. in [Electron. J. Combin. 21(3):#P3.6 (2014) that$$ \\mu_{1}(G) \\mu_{1}(\\bar{G}) \\leqslant n(n-1), $$where $\\bar{G}$ is the complement of $G$, and equality holds if and only if $G$ or $\\bar{G}$ is isomorphic to the join of an isolated vertex and a disconnected graph of order $n-1$.\r\nThey proved that this conjecture holds for bipartite graphs. In this paper, we completely confirm this conjecture. Furthermore, we propose a more general conjecture that for any graph $G$ with $n$ vertices and $k \\leq \\frac{3n}{4}$,$$ \\mu_k(G) \\mu_k(\\bar{G})\\leq n(n-k), $$and equality holds if and only if $G$ or $\\bar{G}$ is isomorphic to the join of $K_{k}$ and a disconnected graph on $n-k$ vertices and has at least $k+1$ connected components.\r\nWe also prove that it is true for $\\frac{n}{2}\\leq k \\leq \\frac{3n}{4}$, and for each $k \\geq \\frac{3n}{4}+1$, a counterexample is given.<\/jats:p>","DOI":"10.37236\/14060","type":"journal-article","created":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T17:13:34Z","timestamp":1762190014000},"source":"Crossref","is-referenced-by-count":0,"title":["A Conjecture on the Nordhaus-Gaddum Product Type Inequality for Laplacian Eigenvalue of a Graph"],"prefix":"10.37236","volume":"32","author":[{"given":"Qi","family":"Chen","sequence":"first","affiliation":[]},{"given":"Ji-Ming","family":"Guo","sequence":"additional","affiliation":[]},{"given":"Wen-Jun","family":"Li","sequence":"additional","affiliation":[]},{"given":"Zhiwen","family":"Wang","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2025,11,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i4p37\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i4p37\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T17:13:34Z","timestamp":1762190014000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i4p37"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,3]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,10,3]]}},"URL":"https:\/\/doi.org\/10.37236\/14060","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2025,11,3]]},"article-number":"P4.37"}}