Existence of a finite group whose subgroup generating bipartite graph violates the Hansen–Vukičević inequality

Determine whether there exists a finite group G for which the subgroup generating bipartite graph B(G)—with vertex set G × G union L(G) and an edge between (a, b) and H whenever H = ⟨a, b⟩—fails to satisfy the Hansen–Vukičević inequality M2(B(G))/|E(B(G))| ≤ M1(B(G))/|V(B(G))| comparing the first and second Zagreb indices.

Background

The paper studies the subgroup generating bipartite graph B(G) associated to a finite group G, where vertices are partitioned into G × G and the subgroup lattice L(G), with adjacency defined by generation. It derives closed-form expressions for the first and second Zagreb indices M1(B(G)) and M2(B(G)) via probabilities Pr_H(G) that a random pair generates a subgroup H ≤ G.

The authors establish that B(G) satisfies the Hansen–Vukičević inequality M2/|E| ≤ M1/|V| for several large families of groups: cyclic groups of orders 2p, 2p^2, 4p, 4p^2, and p^n; dihedral groups of orders 2p and 2p^2; and dicyclic groups of orders 4p and 4p^2 (for primes p). They report no counterexample among these cases and thus pose the problem of whether any finite group yields a violation for B(G).