Computing dominating number and dominant metric dimension for zero divisor graphs of order at most 10 of small finite commutative rings

Nasir Ali; Maysoon  Qousini; Mubarrah Tariq; Sehrish  Riaz; Muhammad Imran  Qureshi; Hafiz Muhammad Afzal Siddiqui

doi:10.5269/bspm.76442

Nasir Ali COMSATS University Islamabad, Vehari Campus
Maysoon Qousini Al-Zaytoonah University of Jordan, Faculty of Science and Information Technology, Amman, Jordan - 11183.
Mubarrah Tariq Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Pakistan.
Sehrish Riaz Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Pakistan.
Muhammad Imran Qureshi Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Pakistan.
Hafiz Muhammad Afzal Siddiqui Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan.

Resumo

In this article we compute the dominating number (DN) and dominant metric dimension (Ddim) of zero divisor graphs of some small finite commutative rings with order not exceeding 14. Consider a commutative ring denoted as and let represent its zero-divisor graph (ZD-graph). The vertices of these graphs correspond to the non-zero divisors (ZD) within the commutative ring (CR), where an edge connects two distinct vertices if their product in the ring results in zero. This paper focuses on studying the domination number and dominant metric dimension for zero divisor graphs of orders 3, 4, 5, 6, 7, 8, 9, and 10 within a small finite commutative ring with a unity. Employing a combination of computational methods and mathematical techniques, our research sheds light on the structural nuances of these small commutative rings, enhancing our comprehension of their algebraic behavior and paving the way for potential applications in algebraic theory and related fields.

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