Zagreb indices of subgroup generating bipartite graph
Shrabani Das, Ahmad Erfanian, Rajat Kanti Nath
TL;DR
This work studies Zagreb-type and related topological indices for the subgroup generating bipartite graph $\mathcal{B}(G)$ of a finite group $G$, connecting graph structure to subgroup-generation probabilities via $\deg_{\mathcal{B}(G)}(H)=|G|^2\Pr_H(G)$. It derives compact expressions $M_1(\mathcal{B}(G))=|G|^2+|G|^4\sum_{H}(\Pr_H(G))^2$ and $M_2(\mathcal{B}(G))=M_1(\mathcal{B}(G))- |G|^2$, and establishes a necessary-and-sufficient criterion for Hansen–Vukičević conjecture in terms of $P=\sum_H(\Pr_H(G))^2$ as $|L(G)|P-1\ge0$. The conjecture is verified for several group families (cyclic of orders $2p,2p^2,4p,4p^2,p^n$; dihedral $D_{2p},D_{2p^2}$; dicyclic $Q_{4p},Q_{4p^2}$), with explicit $M_1,M_2$ values and positivity checks, while also computing Randic, ABC, GA, Harmonic, and SCI indices from the same subgroup-structure data. The results reinforce HV conjecture for these classes and provide closed-form topological indices for $\mathcal{B}(G)$ across multiple group families, illustrating a tight integration between group-subgroup structure and graph-theoretic invariants.
Abstract
Let $G$ be a group and $L(G)$ be the set of all subgroups of $G$. The subgroup generating bipartite graph $\mathcal{B}(G)$ defined on $G$ is a bipartite graph whose vertex set is partitioned into two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in L(G)$ are adjacent if $H$ is generated by $a$ and $b$. In this paper, we deduce expressions for first and second Zagreb indices of $\mathcal{B}(G)$ and obtain a condition such that $\mathcal{B}(G)$ satisfy Hansen-Vuki{č}evi{ć} conjecture [Hansen, P. and Vuki{č}evi{ć}, D. Comparing the Zagreb indices, {\em Croatica Chemica Acta}, \textbf{80}(2), 165-168, 2007]. It is shown that $\mathcal{B}(G)$ satisfies Hansen-Vuki{č}evi{ć} conjecture if $G$ is a cyclic group of order $2p, 2p^2, 4p$, $4p^2$ and $p^n$; dihedral group of order $2p$ and $2p^2$; and dicyclic group of order $4p$ and $4p^2$ for any prime $p$. While computing Zagreb indices of $\mathcal{B}(G)$ we have computed $°_{\mathcal{B}(G)}(H)$ for all $H \in L(G)$ for the above mentioned groups. Using these information we also compute Randic Connectivity index, Atom-Bond Connectivity index, Geometric-Arithmetic index, Harmonic index and Sum-Connectivity index of $\mathcal{B}(G)$.