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In this paper, a novel eccentric neighborhood degree-based topological indices, termed eccentric neighborhood Zagreb indices, have been conceptualized and its discriminating power investigated with regard to the predictability of the boiling point of the chemical substances. The discriminating power of the eccentric neighborhood Zagreb indices was compared with that of Wiener and eccentric connectivity indices. Some explicit results for those new indices for some graphs and graph operations such as join, disjunction, composition, and symmetric difference.