A Spectral Tur\'an Problem for a Fixed Tree

TL;DR

This paper investigates the spectral Turán problem for trees, providing bounds on spectral extremal functions by parametrizing trees based on bipartition and minimum degree, and employing spectral and embedding techniques.

Contribution

It introduces a new parametrization of trees for spectral Turán problems and derives bounds using spectral and embedding methods, extending recent results on the spectral Erdős–Sós conjecture.

Findings

Bounds on spectral extremal functions with error terms of Θ(n^{-1/2}) and Θ(n^{-1})

Characterization of spectral extremal graphs for fixed trees based on bipartition parameters

New embedding results for trees into specific graph constructions

Abstract

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 $T \in \mathcal{T}_{m,l+1}^{\delta}$ if $T$ is a tree of order $m$, where the order of the smaller partite set $A$ of $T$ is $l+1$, and $\delta$ is the minimum degree of the vertices in $A$. The motivation for this parametrization comes from the recent proof of the spectral Erd\H{o}s-S\'os conjecture. For a given fixed tree $T$, we describe $\mathrm{SPEX}(n,T)$ and consequently, bound $\mathrm{spex}(n,T)$ in terms of $m,l,\delta$ for that tree. Our approach combines spectral arguments with new results and constructions on embedding a tree $T \in \mathcal{T}_{m,l+1}^{\delta}$ into graphs of the form $\overline{K}_l \vee m S_{\delta}$. We give bounds on $\mathrm{spex}(n,T)$ within an error of $\Theta(n^{-1/2})$ and $\Theta(n^{-1})$ that are based on our embedding results for the given $T$.