The inequality λ₁(G)^r ≤ ∑ w_r(v) ⋅ (c_G(v)−1)/c_G(v) holds for every finite simple graph G and r ≥ 1, confirming the Kannan-Kumar-Pragada conjecture.
Seneta,Non-negative Matrices and Markov Chains, Springer Series in Statistics, Springer, New York
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A local Tur\'an inequality for walks and the spectral radius
The inequality λ₁(G)^r ≤ ∑ w_r(v) ⋅ (c_G(v)−1)/c_G(v) holds for every finite simple graph G and r ≥ 1, confirming the Kannan-Kumar-Pragada conjecture.