On extremal values of some degree-based topological indices with a forbidden or a prescribed subgraph

Xu in 2011 determined the largest value of the second Zagreb index in an n-vertex graph G with clique number k, and also the smallest value with the additional assumption that G is connected. We extend these results to other degree-based topological indices. The key property of the clique number in the first result is that G is Kk+1-free, while the key property in the second result is that G contains a Kk. We also extend our investigations to other forbidden/prescribed subgraphs. Our main tool is showing that several degree-based topological indices are equal to the weighted sum of the number of some subgraphs of G. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).