Turan's theorem implies Stanley's bound

Let G be a graph with m edges and let ρ be the largest eigenvalue of its adjacency matrix. It is shown that ρ≤2(1-⌊ 1/2+2m+1/4 ⌋-1)m, \rho \le \sqrt {2\left( {1 - {{\left\lfloor {1/2 + \sqrt {2m + 1/4} } \right\rfloor }^{ - 1}}} \right)} m, improving the well-known bound of Stanley. Moreover, writing ω for the clique number of G and Wk for the number of its walks on k vertices, it is shown that the sequence { ((1-1/ω)W2k)1/2k } k=1∞ \left\{ {{{\left( {\left( {1 - 1/\omega } \right){W_{{2^k}}}} \right)}^{1/{2^k}}}} \right\}_{k = 1}^\infty is nonincreasing and converges to ρ.