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For $r=1$, this is Wilf's inequality; for $r=2$, it gives Nikiforov's spectral Tur\\'an theorem. Recently, Liu and Ning proved local versions of these two inequalities, strengthening both Wilf's inequality and Nikiforov's spectral Tur\\'an theorem. 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For $r=1$, this is Wilf's inequality; for $r=2$, it gives Nikiforov's spectral Tur\\'an theorem. Recently, Liu and Ning proved local versions of these two inequalities, strengthening both Wilf's inequality and Nikiforov's spectral Tur\\'an theorem. It is natural to ask whether Nikiforov's spectr"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For every finite simple graph G and every integer r ≥ 1, λ₁(G)^r ≤ ∑_{v∈V(G)} w_r(v) (c_G(v)−1)/c_G(v). 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