Turán'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, ρ ≤ &sqrt; {2&left;({1-{&left;⌊ {1/2 + &sqrt; {2m+1/4} &right;⌋-&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/ω &right;){W_{{2k} &right;)}{1/{2k} &right;\{k=1 &infty; is nonincreasing and converges to ρ.

Publication Title

Discussiones Mathematicae - Graph Theory