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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 ..."
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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 ..."
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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.
A spectral stability theorem for large forbidden graphs, submitted for publication. Preprint available at http://arxiv.org/abs/0711.3485
"... Let µ (G) be the largest eigenvalue of a graph G, let Kr (s1,...,sr) be the complete rpartite graph with parts of size s1,...,sr, and let Tr (n) be the rpartite Turán graph of order n. Our main result is: For all r ≥ 2 and all sufficiently small c> 0, ε> 0, every graph G of sufficiently large order ..."
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Cited by 2 (2 self)
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Let µ (G) be the largest eigenvalue of a graph G, let Kr (s1,...,sr) be the complete rpartite graph with parts of size s1,...,sr, and let Tr (n) be the rpartite Turán graph of order n. Our main result is: For all r ≥ 2 and all sufficiently small c> 0, ε> 0, every graph G of sufficiently large order n with µ (G) ≥ (1 − 1/r −�ε)n satisfies one of the � conditions: (a) G contains a Kr+1 ⌊cln n⌋,..., ⌊cln n⌋, n1− √ �� c; (b) G differs from Tr (n) in fewer than � ε 1/4 + c 1/(8r+8) � n 2 edges. In particular, this result strengthens the stability theorem of Erdős and Simonovits.
Turán’s theorem inverted
, 2008
"... Let K + r (s1,...,sr) be the complete rpartite graph with parts of size s1 ≥ 2,s2,...,sr with an edge added to the first part. Letting tr (n) be the number of edges of the rpartite Turán graph of order n, we prove that: (A) For all r ≥ 2 and all sufficiently small ε> 0, every graph of sufficiently ..."
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Let K + r (s1,...,sr) be the complete rpartite graph with parts of size s1 ≥ 2,s2,...,sr with an edge added to the first part. Letting tr (n) be the number of edges of the rpartite Turán graph of order n, we prove that: (A) For all r ≥ 2 and all sufficiently small ε> 0, every graph of sufficiently large order n with tr (n) + 1 edges contains a K + r ⌊cln n⌋,..., ⌊cln n⌋, n 1− √ c (B) For all r ≥ 2, there exists c> 0 such that every graph of sufficiently large order n with tr (n) + 1 edges contains a K + r (⌊cln n⌋,..., ⌊cln n⌋). These assertions extend results of Erdős from 1963. We also give corresponding stability results Keywords: clique; rpartite graph; stability, Turán’s theorem