Extremal graphs with maximum complementary second Zagreb index

Recently, a couple of degree-based topological indices, defined using a geometrical point of view of a graph edge, have attracted significant attention and being extensively investigated. Furtula and Oz [Complementary Topological Indices, MATCH Commun. Math. Comput. Chem.\/ 93 (2025) 247--263] introduced a novel approach for devising ``geometrical'' topological indices and focused special attention on the complementary second Zagreb index as a representation of the introduced approach. In the same paper, they also conjectured the maximal graphs of order n with the maximum complementary second Zagreb index. In this paper, we confirm their conjecture.