On spectral Tur\'an theorems: confirming a conjecture of Guiduli and two problems of Nikiforov

TL;DR

This paper proves four sharp results in spectral Turán theory, confirming conjectures and solving problems related to eigenvalues and extremal graph structures, with implications for spectral thresholds and Turán's theorem.

Contribution

It confirms a spectral dense-neighborhood conjecture, links spectral and edge thresholds, and addresses eigenvalue bounds related to Turán graphs, advancing spectral extremal graph theory.

Findings

Confirmed Guiduli's spectral dense-neighborhood conjecture.

Showed spectral threshold detects Turán's edge threshold.

Established that Nikiforov's eigenvalue bound implies Turán's theorem.

Abstract

Let $G$ be an $n$-vertex graph, and let $\lambda(G)$ and $\lambda_n(G)$ denote the largest and smallest eigenvalues of its adjacency matrix. Write $e(G)$ for the number of edges of $G$, $d(G)=2e(G)/n$ for its average degree, and $T_r(n)$ for the $r$-partite Tur\'an graph on $n$ vertices. We prove four sharp results in spectral Tur\'an theory. First, we confirm Guiduli's spectral dense-neighborhood conjecture (1996) in a stronger form: if $\lambda(G)\ge \lambda(T_r(n))$, then either $G\cong T_r(n)$, or there exists a vertex $v$ such that $\lambda(G[N(v)]) > \lambda(T_{r-1}(d(v)))$. Moreover, when $\lambda(G)>\lambda(T_r(n))$, every vertex attaining the maximum entry in any nonnegative Perron eigenvector of $G$ has this property. Second, we answer a problem of Nikiforov (2009) by showing that the exact Tur\'an edge threshold is detected by the exact spectral threshold: for every $r\ge 2$ and every $n$, $\lambda(G)<\lambda(T_r(n))$, implying $e(G)<e(T_r(n)).$ Our proof also determines the equality cases. Third, we answer another question of Nikiforov (2009) by showing that his least-eigenvalue clique bound \[ \omega(G)\ge 1+\frac{2e(G)}{(n-d(G))(d(G)-\lambda_n(G))} \] does imply the concise form of Tur\'an's theorem. Finally, we discuss an open problem proposed by Ai et al. (2026) in \cite{ALNS26+}.