A spectral Erdős-Sós theorem

Abstract

The famous Erdős-Sós conjecture states that every graph of average degree more than t − 1 must contain every tree on t + 1 vertices. In this paper, we study a spectral version of this conjecture. For n > 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 ≥ 2 and sufficiently large n, if a graph on n vertices has adjacency spectral radius at least as large as S n,k and is not isomorphic to S n,k , then it contains all trees on 2k + 2 vertices. Similarly, if a sufficiently large graph has spectral radius at least as large as S + n,k , then it either contains all trees on 2k + 3 vertices or is isomorphic to S + n,k . This answers a two-part conjecture of Nikiforov affirmatively.

An Aα-Spectral Erdős-Sós Theorem

“…In this paper, we consider an A α -spectral version of Erdős-Sós conjecture for α ∈ (0, 1), which extends the main results of Cioabǎ, Desai and Tait [7]. Our main results can be stated as:…”

“…Thus one finds that T − zu + zw is an isomorphic copy of T in G n,k,α , a contradiction. Now, by Claim 13 we see Ĝ n,k,α ∈ G n,k , and so (7) gives a contradiction to the choice of G n,k,α . Therefore, G n,k,α is connected.…”

“…Let k 2 and G be a graph of sufficiently large order n. There are many results involving Conjecture 3, see [21,22,27]. Very recently, Conjecture 3 was completely resolved by Cioabǎ, Desai and Tait [7].…”