Maximizing the Sombor index of unicyclic graphs of fixed order with k pendant vertices

The Sombor index of a graph G is defined as, where dG(u) represents the degree of vertex u.In this paper, we establish a sharp upper bound for the Sombor index of unicyclic graphs with n vertices and k pendant vertices, and characterize the extremal graph attaining this bound.Using a case analysis based on the graph structure and an inductive argument, we demonstrate that the maximum Sombor index is achieved by the graph formed by attaching k pendant vertices to a single vertex of an (n -k)-cycle.