Vertex-degree-based (VDB) topological indices assign to each edge a weight depending only on the degrees of its endpoints. We develop a unified linear and geometric framework for studying these indices on graph classes with bounded maximum degree.
By encoding each graph through its edge-type vector, every VDB index becomes a linear functional on a finite dimensional space. The analytic form of the generating function plays no essential role: a VDB index is completely determined by its values on degree pairs, and we show that every such index coincides with one induced by a symmetric polynomial. An explicit construction is given for the Sombor index.
We then study discrimination. Strong discrimination is infeasible for VDB indices on several natural graph classes, but a weaker and meaningful notion-weak discrimination-admits a clean linear characterization in terms of the difference space induced by edge-type variations. This leads to general criteria based on support restrictions and linear independence of the weights over \( \mathbb{Q} \), covering forbidden edge types as well as other structural constraints.
Finally, we investigate extremal values using the feasible polytope generated by edge-type vectors. Maximizers and minimizers of a VDB index correspond to normal cones of this polytope, providing a geometric explanation for the frequent appearance of stars and paths as extremal trees. Several explicit examples illustrate the framework.