The Szeged Index of Power Graph of Finite Groups

TL;DR

The paper addresses the Szeged index of power graphs of finite groups by first establishing a formula for the Szeged index of generalized joins of graphs, enabling a decomposition into component graphs. It then applies this framework to the power graph of the cyclic group $\mathbb{Z}_n$, describing its structure via the divisor lattice and giving an explicit Szeged-index formula that specializes to several cases, including prime-power and semiprime $n$. A key result relates the Szeged index of the power graph of the dihedral group $\mathrm{D}_n$ to that of $\mathbb{Z}_n^*$ by an additive correction, with a closed-form in the $n=pq$ case. The work is complemented by SAGE code for computing and visualizing the Szeged index of these power graphs, providing a practical tool for researchers studying graph invariants in algebraic structures.

Abstract

The Szeged index of a graph is an invariant with several applications in chemistry. The power graph of a finite group $G$ is a graph having vertex set as $G$ in which two vertices $u$ and $v$ are adjacent if $v=u^m$ or $u=v^n$ for some $m,n\in \mathbb{N}$. In this paper, we first obtain a formula for the Szeged index of the generalized join of graphs. As an application, we obtain the Szeged index of the power graph of the finite cyclic group $\mathbb Z_n$ for any $n>2$. We further obtain a relation between the Szeged index of the power graph of $\mathbb Z_n$ and the Szeged index of the power graph of the dihedral group $\mathrm{D}_n$. We also provide SAGE codes for evaluating the Szeged index of the power graph of $\mathbb{Z}_n$ and $\mathrm{D}_n$ at the end of this paper.

Figures (3)

• Figure 1: $\mathcal{G}=P_3$ for $\mathcal{P}(\mathbb{Z}_{pq})$
• Figure 2: $\mathcal{G}$ for $\mathcal{P}(\mathbb{Z}_{pq^2})$
• Figure 3: $\mathcal{P}(\mathrm D_{6})$

Theorems & Definitions (17)

• Definition 2.1
• Theorem 2.1
• proof
• Theorem 3.1
• Theorem 3.2
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof