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A Spectral Turán Problem for a Fixed Tree

Dheer Noal Desai, Hemanshu Kaul, Bahareh Kudarzi

TL;DR

This work advances the spectral Turán theory for fixed trees by parameterizing trees via a unique bipartition $(A,B)$ with $|A|=l+1$ and minimum degree $\delta$ on $A$, encoded as $T\in\mathcal{T}_{m,l+1}^{\delta}$. The authors develop a framework based on embedding a fixed tree into structured join graphs of the form $K_l$ join $H_2$ (and more generally $\overline{K_l}$ join $m S_{\delta}$), combining combinatorial embeddings with spectral analysis to bound the spectral extremal number $\mathrm{spex}(n,T)$. They prove general bounds $f(\delta-2,n) \le \mathrm{spex}(n,T) \le f(\delta-1,n)$ (where $f(d,n)$ is defined in the paper) with errors such as $\Theta(n^{-1/2})$ or $\Theta(n^{-1})$, and derive sharper $\Theta(n^{-1})$-type bounds when $T$ embeds into $\overline{K_l}$ join $m S_{\delta}$, yielding $\mathrm{spex}(n,T)\le f(\delta-2,n) + O(n^{-1})$. The results also identify structural properties of spectral extremal graphs, showing that for large $n$, $G\in\mathrm{SPEX}(n,T)$ essentially has a join form $K_l\ join H_2$ with tightly controlled $H_2$ and degree distribution; these findings generalize known spectral Erdős–Sós phenomena for paths and connect to broader extremal questions in the spectral setting.

Abstract

We study the spectral Turán problem for trees. To avoid limiting our perspective to specific families of trees, we parametrize trees in terms of their unique bipartition. We say $T \in \mathcal{T}_{m,l+1}^δ$ 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 Erdős-Sós conjecture. For a given fixed tree $T$, we describe $\mathrm{SPEX}(n,T)$ and consequently, bound $\mathrm{spex}(n,T)$ in terms of $m,l,δ$ for that tree. Our approach combines spectral arguments with new results and constructions on embedding a tree $T \in \mathcal{T}_{m,l+1}^δ$ into graphs of the form $\overline{K}_l \vee m S_δ$. We give bounds on $\mathrm{spex}(n,T)$ within an error of $Θ(n^{-1/2})$ and $Θ(n^{-1})$ that are based on our embedding results for the given $T$.

A Spectral Turán Problem for a Fixed Tree

TL;DR

This work advances the spectral Turán theory for fixed trees by parameterizing trees via a unique bipartition with and minimum degree on , encoded as . The authors develop a framework based on embedding a fixed tree into structured join graphs of the form join (and more generally join ), combining combinatorial embeddings with spectral analysis to bound the spectral extremal number . They prove general bounds (where is defined in the paper) with errors such as or , and derive sharper -type bounds when embeds into join , yielding . The results also identify structural properties of spectral extremal graphs, showing that for large , essentially has a join form with tightly controlled and degree distribution; these findings generalize known spectral Erdős–Sós phenomena for paths and connect to broader extremal questions in the spectral setting.

Abstract

We study the spectral Turán problem for trees. To avoid limiting our perspective to specific families of trees, we parametrize trees in terms of their unique bipartition. We say if is a tree of order , where the order of the smaller partite set of is , and is the minimum degree of the vertices in . The motivation for this parametrization comes from the recent proof of the spectral Erdős-Sós conjecture. For a given fixed tree , we describe and consequently, bound in terms of for that tree. Our approach combines spectral arguments with new results and constructions on embedding a tree into graphs of the form . We give bounds on within an error of and that are based on our embedding results for the given .

Paper Structure

This paper contains 13 sections, 25 theorems, 70 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Spectral extremal problems for trees and our motivation
  3. A parametrization of trees and our problem
  4. Outline and discussion of our results
  5. Basic terminology and notation
  6. Tree embedding results and constructions
  7. Proof of Theorem \ref{['thm: max degrees is G[R]']}
  8. Proof of Theorem \ref{['thm: tree embedding']}
  9. Trees embeddable in Tree embeddable
  10. On the structure of spectral extremal graphs
  11. Spectral extremal theorems for a fixed tree
  12. Proof of Theorem \ref{['thm: bounds for T - spex']}
  13. Proof of Theorem \ref{['thm: T-spex tighter bounds version']}

Key Result

Theorem 1

Let $k \geq 2$ and $G$ be an $n$-vertex graph with sufficiently large $n$.

Figures (7)

  • Figure 1: An illustration of the procedure for finding $J'$.
  • Figure 2: An example of a tree $T$ and the corresponding subtree $T^{J'}$
  • Figure 3: A rooted tree $T^I$ with the given vertex $v$ and the corresponding subtree $T^I_v$
  • Figure 4: An example illustrating the procedure in the proof of Lemma \ref{["lem: there is I for T'"]}
  • Figure 5: Setup for Example \ref{['ex for embedding']}
  • ...and 2 more figures

Theorems & Definitions (65)

  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Theorem 2
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Remark 1.3
  • ...and 55 more