Spectral saturation: Inverting the spectral Turán theorem

Abstract

Let μ(G) be the largest eigenvalue of a graph G and Tr (n) be the r-partite Turán graph of order n. We prove that if G is a graph of order n with μ (G) > μ(Tr. (n)), then G contains various large supergraphs of the complete graph of order r + 1, e.g., the complete r-partite graph with all parts of size log'n with an edge added to the first part. We also give corresponding stability results.

Publication Title

Electronic Journal of Combinatorics

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