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

Let mu(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 Erdos-Sos Conjecture that any tree of order t is contained in a graph of average degree greater than t - 2. Let S-n,S-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 Sn,k by adding a single edge joining two vertices of the independent set of Sn,k. In 2010, Nikiforov conjectured that for a given integer k, every graph G of sufficiently large order n with mu(G) >= mu(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 mu(G) >= mu(S-n,S-k) and G not equal S-n,S-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.(c) 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).