Furtula–Oz conjecture on extremal complementary second Zagreb-index graphs

Establish or refute that, for every integer n\geq 5, any connected graph G^* of order n maximizing the complementary second Zagreb index cM_2 is isomorphic to K_k+\overline{K}_{n-k} for some integer k satisfying k<\lceil n/2\rceil.

Background

The complementary second Zagreb index of a graph G is defined by cM_2(G)=\sum_{uv\in E(G)}|d_u(G)^2-d_v(G)^2|. The cited conjecture concerns the structure of a connected order-n graph G^* maximizing this index among all connected graphs of the same order.

The proposed extremal graph is the join K_k+\overline{K}_{n-k}, consisting of a k-vertex clique joined to an independent set of n-k vertices, with k<\lceil n/2\rceil. The present paper proves that an extremal graph has maximum degree n-1, has no edge joining two minimum-degree vertices, and satisfies an upper bound on the number of maximum-degree vertices; it also verifies supporting inequalities for specified bidegreed and tridegreed graph classes, but does not establish the conjecture for all connected graphs.