Furtula–Oz conjecture on the extremal complementary second Zagreb index graph

Let $G$ be a connected graph of order $n$, and let

$$cM_2(G)=\sum_{uv\in E(G)}\left|(d_u(G))^2-(d_v(G))^2\right|,$$

where $d_u(G)$ is the degree of vertex $u$. Let $G^*$ be a graph attaining the maximum value of $cM_2$ among all connected graphs of order $n$. For disjoint graphs $H_1$ and $H_2$, their join $H_1+H_2$ is obtained by adding every edge between $V(H_1)$ and $V(H_2)$; $K_k$ denotes the complete graph of order $k$, and $\overline{K}_{n-k}$ the complement of the complete graph of order $n-k$. Furtula–Oz's conjecture. If $n\geq 5$, then $G^*$ is isomorphic to

$$K_k+\overline{K}_{n-k}$$

for some $k$ satisfying $k<\lceil n/2\rceil$.