Geodesic
Turán’s Theorem Implies Stanley’s Bound
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 601-605

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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,) improving the well-known bound of Stanley. Moreover, writing ω for the clique number of G and W_k for the number of its walks on k vertices, it is shown that the sequence {((1-1/ω)W_2^k)^1/2^k}_k=1^∞ is nonincreasing and converges to ρ.
Keywords: graph spectral radius, Stanley’s bound, Turán’s theorem, clique number, Motzkin-Straus’s inequality, walks 
@article{DMGT_2020_40_2_a14,
     author = {Nikiforov, V.},
     title = {Tur\'an{\textquoteright}s {Theorem} {Implies} {Stanley{\textquoteright}s} {Bound}},
     journal = {Discussiones Mathematicae. Graph Theory},
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     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2020_40_2_a14/}
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Nikiforov, V. Turán’s Theorem Implies Stanley’s Bound. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 601-605. http://geodesic.mathdoc.fr/item/DMGT_2020_40_2_a14/