Metric dimension and Zagreb indices of essential ideal graph of a finite commutative ring
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914857476947968 |
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| author | P, Jamsheena A V, Chithra |
| author_facet | P, Jamsheena A V, Chithra |
| contents | 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 \textit{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.
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 $\mathcal{E}_{\mathbb Z_n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02938 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metric dimension and Zagreb indices of essential ideal graph of a finite commutative ring P, Jamsheena A V, Chithra Combinatorics 05C07, 05C12, 05C25 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 \textit{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. 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 $\mathcal{E}_{\mathbb Z_n}$. |
| title | Metric dimension and Zagreb indices of essential ideal graph of a finite commutative ring |
| topic | Combinatorics 05C07, 05C12, 05C25 |
| url | https://arxiv.org/abs/2407.02938 |