A note on the Independent domination polynomial of zero divisor graph of rings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913186588917760 |
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| author | Rather, Bilal Ahmad |
| author_facet | Rather, Bilal Ahmad |
| contents | In this note we consider the independent domination polynomial problem along with their unimodal and log-concave properties which were earlier studied by Gürsoy, Ülker and Gürsoy (Soft Comp. 2022). We show that the independent domination polynomial of zero divisor graphs of $\mathbb{Z}_{n}$ for $n\in \{ pq, p^{2}q, pqr, p^α\}$ where $p,q,r$ are primes with $2<p<q<r$ are not unimodal thereby contradicting the main result of Gürsoy, Ülker and Gürsoy \cite{gursoy}. Besides the authors show that the zero of the independent domination polynomial of these graphs have only real zero and used concept of Newton's inequalities to establish the log-concave property for the afore said polynomials. We show that these polynomials have complex zeros and the technique of Newton's inequalities are not applicable. Finally, by definition of log-concave, we prove that these polynomials are log-concave and fix the flaws in Theorem 10 of Gürsoy, Ülker and Gürsoy \cite{gursoy}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_02559 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on the Independent domination polynomial of zero divisor graph of rings Rather, Bilal Ahmad Combinatorics Discrete Mathematics 05C25, 05C31, 05C69, 05C90 In this note we consider the independent domination polynomial problem along with their unimodal and log-concave properties which were earlier studied by Gürsoy, Ülker and Gürsoy (Soft Comp. 2022). We show that the independent domination polynomial of zero divisor graphs of $\mathbb{Z}_{n}$ for $n\in \{ pq, p^{2}q, pqr, p^α\}$ where $p,q,r$ are primes with $2<p<q<r$ are not unimodal thereby contradicting the main result of Gürsoy, Ülker and Gürsoy \cite{gursoy}. Besides the authors show that the zero of the independent domination polynomial of these graphs have only real zero and used concept of Newton's inequalities to establish the log-concave property for the afore said polynomials. We show that these polynomials have complex zeros and the technique of Newton's inequalities are not applicable. Finally, by definition of log-concave, we prove that these polynomials are log-concave and fix the flaws in Theorem 10 of Gürsoy, Ülker and Gürsoy \cite{gursoy}. |
| title | A note on the Independent domination polynomial of zero divisor graph of rings |
| topic | Combinatorics Discrete Mathematics 05C25, 05C31, 05C69, 05C90 |
| url | https://arxiv.org/abs/2401.02559 |