This commentary addresses some minor errors in the literature concerning the upper bounds of the first Zagreb index, both related to the necessary and sufficient conditions for equality in the associated inequalities. To resolve these issues, we provide corrected derivations and clarify their implications for the validity of the bounds. Furthermore, we conduct a review of recent papers that have incorrectly applied these inequalities, identifying common misuses and providing revised conclusions to ensure accuracy. Additionally, to ensure completeness and accessibility, we present more concise and elementary proofs of the underlying inequalities, which simplifies the original approach.