Turán’s theorem inverted, submitted for publication. Preprint available at http://arxiv.org/abs/0707.3394 (0)

by V Nikiforov
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Spectral saturation: inverting the spectral Turán theorem

by Vladimir Nikiforov , 2009
"... 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 ..."
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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.