On a Conjecture Concerning the Complementary Second Zagreb Index

The complementary second Zagreb index of a graph G is defined as cM2where du(G) denotes the degree of a vertex u in G and E(G) represents the edge set of G. Let G * be a graph having the maximum value of cM2 among all connected graphs of order n.Furtula and Oz [MATCH Commun.Math.Comput.Chem.93 (2025) 247-263] conjectured that G * is the join K k + K n-k of the complete graph K k of order k and the complement K n-k of the complete graph K n-k such that the inequality k < n/2 holds.We prove that (i) the maximum degree of G * is n -1 and (ii) no two vertices of minimum degree in G * are adjacent; both of these results support the aforementioned conjecture.We also prove that the number of vertices of maximum degree in G * , say k, is at most -2 3 n +