On Intersection Graph of Dihedral Group

TL;DR

This paper studies the intersection graph of dihedral groups, analyzing its properties and calculating various topological indices, including the metric dimension and resolving polynomial for specific cases.

Contribution

It introduces new calculations of topological indices and metric properties for intersection graphs of dihedral groups, especially for groups of order twice a prime square.

Findings

Computed Wiener, Hyper-Wiener, Zagreb, Schultz, Gutman, and eccentric connectivity indices.

Determined the metric dimension of the intersection graph of D_{2p^2}.

Derived the resolving polynomial for the intersection graph of D_{2p^2}.

Abstract

Let $G$ be a finite group. The intersection graph of $G$ is a graph whose vertex set is the set of all proper non-trivial subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K \neq \{e\}$, where $e$ is the identity of the group $G$. In this paper, we investigate some properties and exploring some topological indices such as Wiener, Hyper-Wiener, first and second Zagreb, Schultz, Gutman and eccentric connectivity indices of the intersection graph of $D_{2n}$ for $n=p^2$, $p$ is prime. We also find the metric dimension and the resolving polynomial of the intersection graph of $D_{2p^2}$.