Two Sombor index conjectures from a common core-edge principle

We resolve two extremal conjectures for Sombor-type indices of connected graphs with prescribed order and cyclomatic number. First, we prove the conjecture of Réti, Došlić and Ali for the classical Sombor index, removing the pendant-vertex restriction present in the previous general result. Second, we prove Conjecture 2.1.1 of Ali, Gutman, Réti, Albalahi and Hamza for the elliptic Sombor index. Both results follow from a common core-edge principle: after deleting a universal vertex from a maximizer, the problem becomes a local optimization on a graph with exactly the prescribed cyclomatic number of edges. A local unbalancing inequality forces the residual graph to be a star. For the elliptic Sombor index this follows from convexity of a chord-slope function; for the classical Sombor index a quantitative compensation argument is used instead.