{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:11Z","timestamp":1753893791528,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $G$ be a graph and let $\\alpha$ be a real number in $[0,1].$ In 2017, Nikiforov proposed the $A_\\alpha$-matrix for $G$ as $A_{\\alpha}(G)=\\alpha D(G)+(1-\\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of $G$, respectively. The largest eigenvalue of $A_{\\alpha}(G)$ is called the $A_\\alpha$-index of $G.$ The famous Erd\u0151s-S\u00f3s conjecture states that every $n$-vertex graph with more than $\\frac{1}{2}(k-1)n$ edges must contain every tree on $k+1$ vertices. In this paper, we consider an $A_\\alpha$-spectral version of this conjecture. For $n&gt;k,$ let $S_{n,k}$ be the join of a clique on $k$ vertices with an independent set of $n-k$ vertices and denote by $S^+_{n,k}$ the graph obtained from $S_{n,k}$ by adding one edge. We show that for fixed $k\\geq2,\\,0&lt;\\alpha&lt;1$ and $n\\geq\\frac{88k^2(k+1)^2}{\\alpha^4(1-\\alpha)}$, if a graph on $n$ vertices has $A_\\alpha$-index at least as large as $S_{n,k}$ (resp. $S^+_{n,k}$), then it contains all trees on $2k+2$ (resp. $2k+3$) vertices, or it is isomorphic to $S_{n,k}$ (resp. $S^+_{n,k}$). These extend the results of Cioab\u0103, Desai and Tait (2022), in which they confirmed the adjacency spectral version of the Erd\u0151s-S\u00f3s conjecture.<\/jats:p>","DOI":"10.37236\/11593","type":"journal-article","created":{"date-parts":[[2023,9,21]],"date-time":"2023-09-21T18:04:35Z","timestamp":1695319475000},"source":"Crossref","is-referenced-by-count":0,"title":["An A\u03b1-Spectral Erd\u0151s-S\u00f3s Theorem"],"prefix":"10.37236","volume":"30","author":[{"given":"Ming-Zhu","family":"Chen","sequence":"first","affiliation":[]},{"given":"Shuchao","family":"Li","sequence":"additional","affiliation":[]},{"given":"Zhao-Ming","family":"Li","sequence":"additional","affiliation":[]},{"given":"Yuantian","family":"Yu","sequence":"additional","affiliation":[]},{"given":"Xiao-Dong","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2023,9,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i3p34\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i3p34\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,21]],"date-time":"2023-09-21T18:04:36Z","timestamp":1695319476000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v30i3p34"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,22]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,7,14]]}},"URL":"https:\/\/doi.org\/10.37236\/11593","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2023,9,22]]},"article-number":"P3.34"}}