Adjacency Matrix and Energy of the Line Graph of $\Gamma(\mathbb{Z}_n)$

TL;DR

This paper investigates the properties of the line graph of the zero divisor graph of bZ_n, focusing on adjacency matrix, neighborhood, Wiener index, and energy, with specific results for cases where n is a product of two primes or a square of a prime.

Contribution

It provides new insights into the structure and spectral properties of the line graph of zero divisor graphs of bZ_n, including explicit calculations and MATLAB code.

Findings

Determined the adjacency matrix of the line graph for specific n.

Calculated Wiener index and energy for the line graph.

Provided MATLAB code for computations.

Abstract

Let $\Gamma(\mathbb{Z}_n)$ be the zero divisor graph of the commutative ring $\mathbb{Z}_n$ and $L(\Gamma(\mathbb{Z}_n))$ be the line graph of $\Gamma(\mathbb{Z}_n)$. In this paper, we discuss about the neighborhood of a vertex, the neighborhood number and the adjacency matrix of $L(\Gamma(\mathbb{Z}_n))$. We also study Wiener index and energy of $L(\Gamma(\mathbb{Z}_n))$, where $n=pq$, $p^2$ respectively for $p$ and $q$ are primes. Moreover, we give MATLAB coding of our calculations.