Metric dimension and Zagreb indices of essential ideal graph of a finite commutative ring

Let $R$ be a commutative ring with unity. The essential ideal graph ${\mathcal{E}}_R$ of $R$ is a graph whose vertex set consists of all nonzero proper ideals of $\mathit{\text{R}}$. Two vertices $\widehat{I}$ and $\widehat{J}$ are adjacent if and only if $\widehat{I}+\widehat{J}$ is an essential ideal. In this paper, we characterize the graph ${\mathcal{E}}_R$ as having a finite metric dimension. Additionally, we identify that the essential ideal graph and annihilating ideal graph of the ring ${\mathbb{Z}}_n$ are isomorphic whenever $n$ is a product of distinct primes. Also, we estimate the metric dimension of the essential ideal graph of the ring ${\mathbb{Z}}_n$. Furthermore, we determine the topological indices, namely the first and the second Zagreb indices, of ${\mathcal{E}}_{{\mathbb{Z}}_n}$.