The number of independent sets in graphs

The following theorem is proved: Let G be a κ-regular graph with n vertices such that the maximal size of an independent set of the graph G is equal to μ. Then i(G) ≤ (2^{mu log } (1 + tfrac{n}{{2mu }}) + O(nsqrt {k^{ - 1} log k} )). This statement is generalized to the case of quasi-regular graphs. As a corollary, an upper bound for the number of independent sets in extenders is obtained.