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 set of vertex consists of all nonzero proper ideals of R. Two vertices \hat{I} and \hat{J} are adjacent if and only if \hat{I}+ \hat{J} is an essential ideal. In this paper, we characterize the graph \mathcal{E}_{R} as having a finite metric dimension. Furthermore, we identify that the essential ideal graph and the annihilating ideal graph of the ring \mathbb{Z}_{n} are isomorphic whenever n is a product of distinct primes. In addition, we estimate the metric dimension of the essential ideal graph of the ring \mathbb{Z}_{n}. Moreover, we determine the topological indices, namely the first and second Zagreb indices, of \mathcal{E}_{\mathbb Z_n}.

کلیدواژه ها:

Essential ideal graph ، Metric dimension ، first and second Zagreb indices