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₊₁. 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.