A spectral Erd\H{o}s-S\'os theorem

TL;DR

This paper proves a spectral version of the Erdős-Sós conjecture, showing that large graphs with high spectral radius contain all trees of certain sizes, confirming a conjecture by Nikiforov.

Contribution

It establishes spectral conditions guaranteeing the presence of all trees of specific sizes, confirming a conjecture of Nikiforov.

Findings

Graphs with spectral radius at least that of S_{n,k} contain all trees on 2k+2 vertices.

Graphs with spectral radius at least that of S_{n,k}^+ contain all trees on 2k+3 vertices or are isomorphic to S_{n,k}^+.

The results affirm a two-part conjecture of Nikiforov.

Abstract

The famous Erd\H{o}s-S\'os 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 $k\geq 2$ 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.