The spectral radius of graphs without trees of diameter at most four

Abstract

Nikiforov [The spectral radius of graphs without paths and cycles of specified length. Linear Algebra Appl. 2010;432(9):2243–2256] conjectured that for a given integer k, any graph G of sufficiently large order n with spectral radius contains all trees of order 2k+2, unless , where denotes the join of a complete graph of order k and an empty graph of order n−k. In this article, we show that the conjecture is true for trees of diameter at most four, more specifically, we prove that, for and , every graph G of order n with contains all trees T of order 2k+2 and of diameter at most four as a subgraph, unless .