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The graphs having the maximum value of certain bond incident degree indices (including the second Zagreb index, general sum-connectivity index, and general zeroth-order Randić index) in the class of all connected graphs with fixed order and number of pendent vertices are characterized in this paper. The problem of finding graphs having the minimum values of the second Zagreb index and general zeroth-order Randić index from the aforementioned class of connected graphs is also addressed. One of the obtained results about the first Zagreb index has already been proved in the papers [I. Gutman, M. Kamran Jamil, N. Akhter, Trans. Combin. 4 (2015) 43-48] and [M. Enteshari, B. Taeri, MATCH Commun. Math. Comput. Chem. 86 (2021) 17-28]; however, it is proved here by another method with a short proof. Moreover, one of the obtained results concerning the second Zagreb index gives a partial solution to a problem attacked in the aforementiond paper of Enteshari and Taeri.

On Bond Incident Degree Indices of Connected Graphs with Fixed Order and Number of Pendent Vertices