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

TL;DR

This paper explores the structure, topological indices, and spectral properties of subgroup generating bipartite graphs of finite cyclic groups, verifying several conjectures and deriving explicit formulas for various graph invariants.

Contribution

It provides explicit structural descriptions, formulas for Zagreb and other indices, and spectral analyses of these graphs for specific cyclic groups, extending existing conjectures.

Findings

Structures of $eta(G)$ for specific cyclic groups are characterized.

Explicit formulas for Zagreb and other degree-based indices are derived.

Spectral properties and energies of the graphs are computed and conjectures verified.

Abstract

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{\v{c}}evi{\'c} conjecture [Hansen, P. and Vuki{\v{c}}evi{\'c}, 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.