An Aα-Spectral Erdős-Sós Theorem

This is the authors' abstract. We don't add key points for this paper.

Let G be a graph and let α be a real number in $[0,1].$ In 2017, Nikiforov proposed the A_α-matrix for G as Aα(G)=α D(G)+(1-α)A(G), where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of G, respectively. The largest eigenvalue of Aα(G) is called the A_α-index of $G.$ The famous Erdős-Sós conjecture states that every n-vertex graph with more than 1/2(k-1)n edges must contain every tree on $k+1$ vertices. In this paper, we consider an A_α-spectral version of this conjecture. For $n>k,$ let Sn,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 Sn,k by adding one edge. We show that for fixed k≥2,\,0<α<1 and n≥88k²(k+1)²/α⁴(1-α), if a graph on n vertices has A_α-index at least as large as Sn,k (resp. S⁺n,k), then it contains all trees on $2k+2$ (resp. $2k+3$) vertices, or it is isomorphic to Sn,k (resp. S⁺n,k). These extend the results of Cioabă, Desai and Tait (2022), in which they confirmed the adjacency spectral version of the Erdős-Sós conjecture.