Furtula–Oz conjecture on extremal complementary second Zagreb graphs

Let $G$ be a graph with $n$ vertices, and let $cM_{2}(G)$ denote its complementary second Zagreb index, defined by

$$cM_{2}(G)=\sum_{uv\in E(G)}\left|d_{G}(u)^{2}-d_{G}(v)^{2}\right|.$$

Here $K_m\vee\overline{K_{n-m}}$ is the join of the complete graph on $m$ vertices and the edgeless graph on $n-m$ vertices. Furtula–Oz conjecture. The graph with $n$ vertices and the maximum complementary second Zagreb index is isomorphic to $K_m\vee\overline{K_{n-m}}$ for some $1\leq m\leq n-1$. This conjecture identifies the proposed extremal structure for the complementary second Zagreb index among graphs of order $n$.