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

TL;DR

The paper analyzes subgroup generating bipartite graphs $\mathcal{B}(G)$ for cyclic groups of orders $pq$, $p^2q$, and $p^2q^2$, deriving their exact disjoint-star decompositions and using these to obtain closed-form Zagreb indices and a broad set of spectral invariants. It verifies the Hansen–Vukičević conjecture for these families and establishes energy-related properties, showing $\mathcal{B}(G)$ is hypoenergetic and satisfies the E-LE conjecture. By computing spectra, energies, and various degree-based indices, the work links the algebraic structure of finite cyclic groups to rich graph-theoretic invariants, providing explicit formulas and comprehensive characterizations. These results contribute to understanding graphs on groups and their chemical-graph-inspired indices and energy measures, with potential applications in algebraic graph theory and spectral graph analysis.

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{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.

Theorems & Definitions (55)

• Theorem 2.1
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Conjecture 3.1
• Lemma 3.1
• proof