On a conjecture of Nikiforov involving a spectral radius condition for a graph to contain all trees

TL;DR

This paper advances the understanding of spectral radius conditions ensuring the presence of all trees in a graph, confirming Nikiforov's conjecture for specific classes like brooms and spiders, and introduces a new Turán-type result.

Contribution

It partially confirms Nikiforov's spectral radius conjecture for certain tree classes and presents a novel Turán-type theorem of independent interest.

Findings

Confirmed Nikiforov's conjecture for brooms and some spiders.

Developed a new Turán-type result.

Provided spectral radius conditions guaranteeing tree containment.

Abstract

We partly confirm a Brualdi-Solheid-Tur\'{a}n type conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erd\H{o}s-S\'os Conjecture that any tree of order $t$ is contained in a graph of average degree greater than $t-2$. We confirm Nikiforov's Conjecture for all brooms and for a larger class of spiders. For our proofs we also obtain a new Tur\'{a}n type result which might turn out to be of independent interest.