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.