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 $K_{k+1}$-free, while the key property in the second result is that $G$ contains a $K_{k+1}$. 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 $G$.