Certain topological indices and spectral properties of SGB-graphs of finite cyclic groups

Let $L(G)$ be the set of all subgroups of a group $G$. The subgroup generating bipartite graph $\mathcal{B}(G)$ defined on $G$ is a bipartite graph whose vertex set is the union of 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 realize the structures of $\mathcal{B}(G)$ for cyclic groups of order $pq, p^2q$ and $p^2q^2$, where $p$ and $q$ are primes and $p \neq q$. We also deduce expressions for first and second Zagreb indices of these graphs and check the validity of 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]. Expressions of certain other degree-based topological indices of these graphs are also computed. We further compute various spectra and their corresponding energies of $\mathcal{B}(G)$ if $G$ is any cyclic group of order $p^n, pq, p^2q$ and $p^2q^2$, where $p$ and $q$ are two distinct primes and $n \geq 1$. We conclude the paper showing that $\mathcal{B}(G)$ satisfies E-LE conjecture [Gutman, I., Abreu, N. M. M., Vinagre, C. T. M., Bonifacioa, A. S. and Radenkovic, S. Relation between energy and Laplacian energy, {\em MATCH Communications in Mathematical and in Computer Chemistry}, \textbf{59}, 343–354, 2008] for these groups.

💡 Research Summary

The paper studies the Subgroup Generating Bipartite graph (SGB‑graph) (\mathcal{B}(G)) associated with a finite group (G). By definition, the vertex set of (\mathcal{B}(G)) is the disjoint union of the Cartesian product (G\times G) and the set of all subgroups (L(G)); a pair ((a,b)) and a subgroup (H) are adjacent precisely when (H=\langle a,b\rangle). The authors focus on cyclic groups of orders (pq), (p^{2}q) and (p^{2}q^{2}) (with distinct primes (p\neq q)), and also treat the general cyclic group (C_{p^{n}}).

Structural Decomposition.
Using elementary properties of cyclic groups, they show that (\mathcal{B}(G)) decomposes into a disjoint union of star graphs. For example, \