Extremal graphs with maximum complementary second Zagreb index

The complementary second Zagreb index of a graph $ G $ is defined as $ cM_{2}(G) = \sum_{uv \in E(G)} |d_{G}(u)^{2}-d_{G}(v)^{2}| $. In this paper, we prove that a graph having maximum complementary second Zagreb index among all graphs of order $ n $ is isomorphic to $ K_{m} \vee \overline{K_{n-m}} $ for some $ 1 \leq m < \lceil \frac{n}{2} \rceil $, confirming a conjecture of Furtula and Oz.