Abstract
Let \(\mathcal{G}_n\) be the set of graphs with n vertices and
\(\mathcal{H}\subseteq\mathcal{G}_n\). For each \(H \in \mathcal{H}\),
let \(m(H) = \{m_{i,j}(H)\}\), where \(m_{i,j}(H)\) is the number of
edges in \(H\) that join a vertex of degree \(i\) with a vertex of degree
\(j\). A vertex-degree-based (VDB, for short) topological index
\(\varphi\) is discriminating over \(\mathcal{H}\) if non-isomorphic
graphs in \(\mathcal{H}\) have different values of \(\varphi\). We say
that \(\varphi\) is weakly discriminating over \(\mathcal{H}\) if the
following weaker condition is satisfied for every \(H, H' \in \mathcal{H}\):
\[
\varphi(H) = \varphi{H'} \Longleftrightarrow m(H) = m(H')
\]
Let \(\mathcal{CT}_n\) be the set of chemical trees with \(n\) vertices.
In this paper we show that many of the well-known VDB topological indices
are not weakly discriminating over \(\mathcal{CT}_n\). However, the recently
introduced Sombor index is weakly discriminating over \(\mathcal{CT}_n\).
Also, we give conditions under which a VDB topological index \(\varphi\)
is weakly discriminating over an arbitrary class \(\mathcal{H}\subseteq\mathcal{G}_n\).
