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

We partly confirm a Brualdi-Solheid-Turán type conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erdős-Sós 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 Turán type result which might turn out to be of independent interest.