A Spectral Tur\'an Problem for a Fixed Tree

We study the spectral Tur\'an problem for trees. To avoid limiting our perspective to specific families of trees, we parametrize trees in terms of their unique bipartition. We say T ∈ Tm,l+1δ if T is a tree of order m, where the order of the smaller partite set A of T is $l+1$, and δ is the minimum degree of the vertices in A. The motivation for this parametrization comes from the recent proof of the spectral Erd{o}s-S\'os conjecture. For a given fixed tree T, we describe SPEX(n,T) and consequently, bound spex(n,T) in terms of m,l,δ for that tree. Our approach combines spectral arguments with new results and constructions on embedding a tree T ∈ Tm,l+1δ into graphs of the form K̄ₗ m Sδ. We give bounds on spex(n,T) within an error of Θ(n-1/2) and Θ(n⁻¹) that are based on our embedding results for the given T.