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

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.