Cliques and the spectral radius

We prove a number of relations between the number of cliques of a graph G   and the largest eigenvalue μ(G)μ(G) of its adjacency matrix. In particular, writing ks(G)ks(G) for the number of s-cliques of G  , we show that, for all r⩾2r⩾2,μr+1(G)⩽(r+1)kr+1(G)+∑s=2r(s−1)ks(G)μr+1−s(G), and, if G is of order n, thenkr+1(G)⩾(μ(G)n−1+1r)r(r−1)r+1(nr)r+1.