Spectral saturation: inverting the spectral Turan theorem

TL;DR

This paper proves that graphs with a spectral radius exceeding that of the r-partite Turan graph necessarily contain specific supergraphs of complete graphs, completing a longstanding problem and providing stability results.

Contribution

It establishes spectral conditions that guarantee the presence of certain supergraphs, advancing the spectral Turan theorem and resolving Erdős's 1963 conjecture.

Findings

Graphs with spectral radius above Turan threshold contain supergraphs of K_{r+1}

Identifies a complete r-partite graph of size log n with an added edge

Provides stability results related to spectral saturation

Abstract

We prove that if the spectral radius of a graph G of order n is larger than the spectral radius of the r-partite Turan graph of the same order, then G contains various supergraphs of the complete graph of order r+1. In particular G contains a complete r-partite graph of size log n with one edge added to the first part. These results complete a project of Erdos from 1963. We also give corresponding stability results.