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Spectral saturation: inverting the spectral Turán theorem
, 2009
"... Let µ (G) be the largest eigenvalue of a graph G and Tr (n) be the rpartite 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 rpartite graph with all parts of ..."
Abstract

Cited by 4 (3 self)
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Let µ (G) be the largest eigenvalue of a graph G and Tr (n) be the rpartite 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 rpartite graph with all parts of size log n with an edge added to the first part. We also give corresponding stability results.
Graphs with many copies of a given subgraph
"... Let c> 0, and H be a fixed graph of order r. Every graph on n vertices containing at least cnr copies of H contains a “blowup ” of H with r − 1 vertex classes of size ⌊ cr2 ln n ⌋ and one vertex class of size greater than n1−cr−1. A similar result holds for induced copies of H. Main results Suppose ..."
Abstract

Cited by 3 (2 self)
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Let c> 0, and H be a fixed graph of order r. Every graph on n vertices containing at least cnr copies of H contains a “blowup ” of H with r − 1 vertex classes of size ⌊ cr2 ln n ⌋ and one vertex class of size greater than n1−cr−1. A similar result holds for induced copies of H. Main results Suppose that a graph G of order n contains cnr copies of a given subgraph H on r vertices. How large “blowup ” of H must G contain? When H is an rclique, this question was answered in [3]: G contains a complete rpartite graph with r − 1 parts of size ⌊ cr ln n ⌋ and one part larger than n1−cr−1. The aim of this note is to answer this question for any subgraph H. First we define precisely a “blowup ” of a graph: given a graph H of order r and positive integers x1,..., xr, we write H(x1,..., xr) for the graph obtained by replacing each vertex u ∈ V (H) with a set Vu of size xu and each edge uv ∈ E(H) with a complete bipartite graph with vertex classes Vu and Vv.