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

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+}.