Spectral skeletons and applications

TL;DR

This paper characterizes the structure of graphs with maximum spectral radius avoiding certain subgraphs, extending classical extremal graph theory results and providing applications in spectral graph analysis.

Contribution

It provides a structural characterization of extremal graphs with maximum spectral radius avoiding a family of forbidden subgraphs, generalizing previous extremal results.

Findings

Characterization of graphs with maximum spectral radius avoiding specific subgraphs.

Extension of classical extremal graph results to spectral extremal problems.

Applications in spectral graph theory and related fields.

Abstract

For a graph $G$, its spectral radius $\rho(G)$ is the largest eigenvalue of its adjacency matrix. Let $\mathcal{F}$ be a finite family of graphs with $\min_{F\in \mathcal{F}}\chi(F)=r+1\geq3$, where $\chi(F)$ is the chromatic number of $F$. Set $t=\max_{F\in\mathcal{F}}|F|$. Let $T(rt,r)$ be the Tur\'{a}n 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.