  In this article, we discussed the zero-divisor graph of a commutative ring
with identity $\mathbb{F}_p+u\mathbb{F}_p+u^2 \mathbb{F}_p$ where $u^3=0$ and
$p$ is an odd prime. We find the clique number, chromatic number, vertex
connectivity, edge connectivity, diameter and girth of a zero-divisor graph
associated with the ring. We find some of topological indices and the main
parameters of the code derived from the incidence matrix of the zero-divisor
graph $\Gamma(R).$ Also, we find the eigenvalues, energy and spectral radius of
both adjacency and Laplacian matrices of $\Gamma(R).$
