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

Abstract

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 = K k ∨ K 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 K 1,k+1 by subdividing each of its k + 1 edges once.

A spectral Erdős-Sós theorem

“…Partial results towards this conjecture were given in [16,18,19] for trees of diameter at most 4 and other special cases. In this paper we prove Conjecture 1.2.…”

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

“…Recently, Hou et al [10] proved that Conjecture 1.1 (a) holds for all trees of diameter at most four. Liu, Broersma and Wang [12] proved that Conjecture 1.1 (b) holds for all trees of diameter at most four, except for the subdivision of K 1,k+1 in which every edge is subdivided precisely once.…”