Domatic polynomials of $\Gamma(Z(R)):$ the zero-divisor graphs of commutative rings
Resumen
The domatic polynomial $DP(G, x)$ of a graph $G$ is defined as $DP(G,x)= \sum_{j=1}^{d(G)} dp(G,j) x^j$, where $dp(G,j)$ represents the number of domatic partition of $G$ with size $j$. In this paper, we find domatic number and domatic polynomial of $\Gamma(Z_n)$ where $n \in \{2s, s^2, st, s^2t,stu,s^\alpha\}$ for distinct prime numbers $s, t\; \text{and}\; u$ with $\alpha >2$ and their roots. Further, we discuss a characterization on $DP(\Gamma(R),x)$. Finally, we establish that their domatic polynomials possess the properties of log-concavity and unimodality.
Descargas
Citas
Ahmadi, M.R., Nezhad, R. J., Energy and Wiener index of zero-divisor graphs, Iranian Journal of Mathematical Chemistry 2.1 (Special Issue on the Occasion of Mircea V. Diudea’s Sixtieth Birthday), 45-51, (2011).
Akbari, S., Mohammadian, A., On the zero-divisor graph of a commutative ring, Journal of algebra 274(2), 847-855, (2004).
Alikhani, S., Bakhshesh, D., Ghanbari,N.,Counting the number of domatic partition of a graph, arXiv preprint arXiv:2407.00103 (2024).
Alikhani, S., Peng, Y. H., Introduction to domination polynomial of a graph, arXiv preprint arXiv:0905.2251 (2009).
Anderson, D. F., Livingston, P. S., The zero-divisor graph of a commutative ring, Journal of Algebra 217, 434-447, (1999).
Asir, T., Rabikka, V., The Wiener index of the zero-divisor graph of Zn, Discrete Applied Mathematics 319, 461-471, (2022).
Beck, I., Coloring of commutative rings, Journal of algebra 116(1), 208-226, (1988).
Chattopadhyay, S., Patra, K. L., Sahoo, B. K., Laplacian eigenvalues of the zero divisor graph of the ring Zn, Linear Algebra and its applications 584, 267-286, (2020).
Goddard, W., Henning, M. A., Independent domination in graphs, A survey and recent results, Discrete Mathematics 313(7), 839-854, (2013).
Gursoy, N. K., Ulker, A., G¨ursoy, A., Independent domination polynomial of zero-divisor graphs of commutative rings, Soft Computing 26(15), 6989-6997, (2022).
Hardy, G., Littlewood, J. E., Polya, G.,Inequalities, Cambridge university press, (1952).
Haynes, T. W., Hedetniemi, S., Slater, P., Fundamentals of domination in graphs, CRC press, (2013).
Henning, M. A., Total Domination in Graphs, Springer Monographs in Mathematics/Springer, (2013).
Lucas, T.G., The diameter of a zero-divisor graph, J. Algebra 301(1), 174- 193, (2006).
Narah, I., Singh, K. R., On the domatic Polynomial of graphs.
Narah, I., Singh, K. R., A characterization on the domatic polynomials.
Young, M., Adjacency matrices of zero-divisor graphs of integers modulo n, Involve, a Journal of Mathematics 8( 5), 753-761, (2015).
Zelinka, B., Domination in generalized Petersen graphs, Czechoslovak Mathematical Journal, 52(1), 11-16, (2002).
Zelinka, B., Domatic number of a graph and its variants, In Annals of Discrete Mathematics 51, 363-369, (1992).
Zelinka, B., Domatic number and degrees of vertices of a graph, Mathematica Slovaca 33(2), 145-147, (1983).
Zelinka, B., On domatic numbers of graphs, Mathematica Slovaca, 31(1), 91-95, (1981).
Derechos de autor 2025 Boletim da Sociedade Paranaense de Matemática
Esta obra está bajo licencia internacional Creative Commons Reconocimiento 4.0.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
