Bounds and power means for the general Randic index

2015/08/31 by Elphick, Clive, Wocjan, Pawel
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1508.07950

Abstract

We review bounds for the general Randić index, R_α = ∑_{ij ∈ E} (d_i d_j)^α, and use the power mean inequality to prove, for example, that R_α≥ mλ^2α for α< 0, where λ is the spectral radius of a graph. This enables us to strengthen various known lower and upper bounds for R_α and to generalise a non-spectral bound due to Bollobás et al. We also prove that the zeroth-order general Randić index, Q_α= ∑_{i ∈ V} d_i^α≥ nλ^α for α< 0.

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