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In this paper we define a new graph-theoretic cyclicity index \(CW(G)\) as a natural generalization of the global cyclicity index \(C(G)\) when arbitrary resistances are allocated to each edge of an electrical network. Upper and lower bounds for \(CW(G)\) are then provided using a powerful technique, based on p-majorization, which extends our prior studies (Bianchi et al. in Discrete Appl. Math., 2014, doi:10.1016/j.dam.2014.10.037; Bianchi et al. in Math. Inequal. Appl. 16(2):329-347, 2013). These new results on weighted majorization are of interest in themselves and may be applied also in other scenarios.

Bounds for the global cyclicity index of a general network via weighted majorization