Spectral skeletons and applications

TL;DR

The paper investigates spectral-extremal problems for ${\mathcal F}$-free graphs with ${\min}_{F\in{\mathcal F}}{\chi}(F)=r+1$, introducing the SPEX extremal class and a spectral skeleton framework built on symmetric subgraphs and near-Turán structure. It proves that, under a symmetry-based setup, spectral extremal graphs are highly structured and close to Turán graphs, with a collection of supporting lemmas (edge-swap, stability, and balance) driving the conclusions. The main contributions include a detailed balance result, a partition-based description of SPEX graphs, and applications to powers of cycles, yielding explicit unique extremal graphs in several cases. The findings provide a spectral analogue to classical Turán-type results, with implications for identifying extremal graphs via a compact, skeletonized form that is amenable to algorithmic and SEO use. Overall, the work extends spectral Turán theory by characterizing the precise structure of SPEX graphs and demonstrating concrete applications to cycle powers.

Abstract

For a graph $G$, its spectral radius $ρ(G)$ is the largest eigenvalue of its adjacency matrix. Let $\mathcal{F}$ be a finite family of graphs with $\min_{F\in \mathcal{F}}χ(F)=r+1\geq3$, where $χ(F)$ is the chromatic number of $F$. Set $t=\max_{F\in\mathcal{F}}|F|$. Let $T(rt,r)$ be the Turán graph of order $rt$ with $r$ parts. Assume that some $F_{0}\subseteq\mathcal{F}$ is a subgraph of the graph obtained from $T(rt,r)$ by embedding a path or a matching in one part. Let ${\rm EX}(n,\mathcal{F})$ be the set of graphs with the maximum number of edges among all the graphs of order $n$ containing not any $F\in\mathcal{F}$. Simonovits \cite{S1,S2} gave general results on the graphs in ${\rm EX}(n,\mathcal{F})$. Let ${\rm SPEX}(n,\mathcal{F})$ be the set of graphs with the maximum spectral radius among all the graphs of order $n$ containing not any $F\in\mathcal{F}$. Motivated by the work of Simonovits, we characterize the specified structure of the graphs in ${\rm SPEX}(n,\mathcal{F})$ in this paper. Moreover, some applications are also included.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Lemma 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3