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Cliques and the Spectral Radius
 J. Combin. Theory Ser. B
"... We prove a number of relations between the number of cliques of a graph G and the largest eigenvalue µ (G) of its adjacency matrix. In particular, writing ks (G) for the number of scliques of G, we show that, for all r ≥ 2, µ r+1 (G) ≤ (r + 1) kr+1 (G) + and, if G is of order n, then µ (G) kr+1 (G ..."
Abstract

Cited by 10 (6 self)
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We prove a number of relations between the number of cliques of a graph G and the largest eigenvalue µ (G) of its adjacency matrix. In particular, writing ks (G) for the number of scliques of G, we show that, for all r ≥ 2, µ r+1 (G) ≤ (r + 1) kr+1 (G) + and, if G is of order n, then µ (G) kr+1 (G) ≥ n r� (s − 1) ks (G) µ r+1−s (G), s=2 1 r (r − 1)
On the minimal density of . . .
, 2008
"... For a fixed ρ ∈ [0, 1], what is (asymptotically) the minimal possible density g3(ρ) of triangles in a graph with edge density ρ? We completely solve this problem by proving that (t − 1) t − 2 g3(ρ) = t(t − ρ(t + 1)) t + � �2 t(t − ρ(t + 1)) t2 (t + 1) 2 where t def ..."
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For a fixed ρ ∈ [0, 1], what is (asymptotically) the minimal possible density g3(ρ) of triangles in a graph with edge density ρ? We completely solve this problem by proving that (t − 1) t − 2 g3(ρ) = t(t − ρ(t + 1)) t + � �2 t(t − ρ(t + 1)) t2 (t + 1) 2 where t def
Joints in graphs
, 2008
"... In 1969 Erdős proved that if r ≥ 2 and n> n0 (r) , every graph G of order n and e (G)> tr (n) has an edge that is contained in at least n r−1 /(10r) 6r (r + 1)cliques. In this note we improve this bound to n r−1 /r r+5. We also prove a corresponding stability result. ..."
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In 1969 Erdős proved that if r ≥ 2 and n> n0 (r) , every graph G of order n and e (G)> tr (n) has an edge that is contained in at least n r−1 /(10r) 6r (r + 1)cliques. In this note we improve this bound to n r−1 /r r+5. We also prove a corresponding stability result.