Cliques and the spectral radius

We prove a number of relations between the number of cliques of a graph G and the largest eigenvalue (G) of its adjacency matrix. In particular, writing ks (G) for the number of s-cliques of G, we show that, for all r 2; r+1 (G) (r + 1) kr+1 (G) + rX s=2 (s 1) ks (G) r+1 s (G) ; and, if G is of order n; then kr+1 (G) (G) n 1 + 1 r r (r 1) r + 1 n r r+1 :

• Type: Journal
• Year: 2007
• Where: JCT
• Authors: Béla Bollobás, Vladimir Nikiforov
• Keywords: Adjacency Matrix, Cliques, JCT 2007, Largest Eigenvalue