{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:42:51Z","timestamp":1753893771743,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $T$ be a $k$-tree equipped with a weighting function $w: V(T)\\cup E(T)\\rightarrow C$, where $k\\geq 3$. The weighted matching polynomial of the weighted $k$-tree $(T,w)$ is defined to be $$ \\mu(T,w,x)= \\sum_{M \\in \\mathcal{M}(T)}(-1)^{|M|}\\prod_{e \\in E(M)}\\mathbf{w}(e)^k \\prod_{v \\in V(T)\\backslash V(M)}(x-w(v)),$$ where $\\mathcal{M}(T)$ denotes the set of matchings (including empty set) of $T$. In this paper, we investigate the eigenvalues of the adjacency tensor $\\mathcal{A}(T,w)$ of the weighted $k$-tree $(T,w)$. The main result provides that $w(v)$ is an eigenvalue of $\\mathcal{A}(T,w)$ for every $v\\in V(T)$, and if $\\lambda\\neq w(v)$ for every $v\\in V(T)$, then $\\lambda$ is an eigenvalue of $\\mathcal{A}(T,w)$ if and only if there exists a subtree $T'$ of $T$ such that $\\lambda$ is a root of $\\mu(T',w,x)$. Moreover, the spectral radius of $\\mathcal{A}(T,w)$ is equal to the largest root of $\\mu(T,w,x)$ when $w$ is real and nonnegative. The result extends a work by Clark and Cooper (On the adjacency spectra of hypertrees, Electron. J. Combin., 25 (2)(2018) $\\#$P2.48) to weighted $k$-trees. As applications, two analogues of the above work for the Laplacian and the signless Laplacian tensors of $k$-trees are obtained.<\/jats:p>","DOI":"10.37236\/10942","type":"journal-article","created":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T07:20:41Z","timestamp":1654240841000},"source":"Crossref","is-referenced-by-count":1,"title":["Spectra of Weighted Uniform Hypertrees"],"prefix":"10.37236","volume":"29","author":[{"given":"Jiang-Chao","family":"Wan","sequence":"first","affiliation":[]},{"given":"Yi","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Fu-Tao","family":"Hu","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2022,6,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i2p40\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i2p40\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T07:20:42Z","timestamp":1654240842000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v29i2p40"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,3]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,4,8]]}},"URL":"https:\/\/doi.org\/10.37236\/10942","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2022,6,3]]},"article-number":"P2.40"}}