An Aα-Spectral Erdos-Sos Theorem

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 Erdos-Sos 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(alpha)-spectral version of this conjecture. For n > k, let S-n,S-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,S-k by adding one edge. We show that for fixed k >= 2, 0 < alpha < 1 and n >= 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,S-k (resp. S-n,k(+)), then it contains all trees on 2k +2 (resp. 2k +3) vertices, or it is isomorphic to S-n,S-k (resp. S-n,k(+)). These extend the results of Cioaba, Desai and Tait (2022), in which they confirmed the adjacency spectral version of the Erdos-Sos conjecture.