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Several members of the class of (exponential) bond incident degree (BID) indices have been recently shown to possess escalating or de-escalating properties, which then resulted in their significant and useful applications. In the present note, we propose a simple equivalent method to determine whether a BID index is escalating or de-escalating. In addition, we present a sufficient condition for an exponential BID index to be escalating or de-escalating. As corollaries, we deduce a number of previously established results (including the main results of Chen et al. (2022), Wei and Liu (2023), and state a few new. Moreover, by solving a conjecture from Eliasi (2022), we determine the unique maximum extremal graph w.r.t. exponential second Zagreb index, among all c-cyclic graphs of order n for 0≤c≤n−2, thus extending the main results of Cruz et al. (2021), Das et al. (2021), Eliasi (2022). © 2023 Elsevier B.V.