Extremal graphs with maximum complementary second Zagreb index

The complementary second Zagreb index of a graph $G$ is defined as $c{M}_2(G)=\sum _{uv\in E(G)}|d_G(u{)}^2-d_G(v{)}^2|$. 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 $1\le m<\lceil \frac{n}{2}\rceil$, confirming a conjecture of Furtula and Oz.