Spectral radius conditions for the existence of all subtrees of diameter at most four

Let μ(G) denote the spectral radius of a graph G. We partly confirm a 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. Let S_n,k be the graph obtained by joining every vertex of a complete graph on k vertices to every vertex of an independent set of order n−k, and let S_n,k⁺ be the graph obtained from S_n,k by adding a single edge joining two vertices of the independent set of S_n,k. In 2010, Nikiforov conjectured that for a given integer k, every graph G of sufficiently large order n with μ(G)≥μ(S_n,k⁺) contains all trees of order 2k+3, unless G=S_n,k⁺. We confirm this conjecture for trees with diameter at most four, with one exception. In fact, we prove the following stronger result for k≥8. If a graph G with sufficiently large order n satisfies μ(G)≥μ(S_n,k) and G≠S_n,k, then G contains all trees of order 2k+3 with diameter at most four, except for the tree obtained from a star on k+2 vertices by subdividing each of its k+1 edges once.