ON PROPERTIES OF PRIME IDEAL GRAPHS OF COMMUTATIVE RINGS
The prime ideal graph of in a finite commutative ring with unity, denoted by , is a graph with elements of as its vertices and two elements in are adjacent if their product is in . In this paper, we explore some interesting properties of . We determined some properties of such as radius, diameter, degree of vertex, girth, clique number, chromatic number, independence number, and domination number. In addition to these properties, we study dimensions of prime ideal graphs, including metric dimension, local metric dimension, and partition dimension; furthermore, we examined topological indices such as atom bond connectivity index, Balaban index, Szeged index, and edge-Szeged index.
R. Diestel, Graph Theory (5th Edition). Springer, 2017.
J. Hamm and A. Way, “PARAMETERS OF THE COPRIME GRAPH OF A GROUP,” International Journal of Group Theory, vol. 10, no. 3, pp. 137–147, Sep. 2021, doi: 10.22108/ijgt.2020.112121.1489.
F. Mansoori, A. Erfanian, and B. Tolue, “Non-coprime graph of a finite group,” in AIP Conference Proceedings, American Institute of Physics Inc., Jun. 2016. doi: 10.1063/1.4954605.
E. Y. Asmarani, A. G. Syarifudin, I. G. A. W. Wardhana, and N. W. Switrayni, “The Power Graph of a Dihedral Group,” EIGEN MATHEMATICS JOURNAL, pp. 80–85, Jan. 2022, doi: 10.29303/emj.v4i2.117.
N. Nurhabibah, A. G. Syarifudin, I. G. A. W. Wardhana, and Q. Aini, “The Intersection Graph of a Dihedral Group,” EIGEN MATHEMATICS JOURNAL, pp. 68–73, Dec. 2021, doi: 10.29303/emj.v4i2.119.
A. Mukhtar, R. Murtaza, S. U. Rehman, S. Usman, and A. Q. Baig, “Computing the size of zero divisor graphs,” Journal of Information and Optimization Sciences, vol. 41, no. 4, pp. 855–864, May 2020, doi: 10.1080/02522667.2020.1745378.
T. C. Burness and E. Covato, “On the prime graph of simple groups,” Bull Aust Math Soc, vol. 91, no. 2, pp. 227–240, Apr. 2015, doi: 10.1017/S0004972714000707.
S. Humaira, P. Astuti, I. Muchtadi-Alamsyah, and A. Erfanian, “The matrix Jacobson graph of finite commutative rings,” Electronic Journal of Graph Theory and Applications, vol. 10, no. 1, pp. 181–197, 2022, doi: 10.5614/ejgta.2022.10.1.12.
K. Nozari and S. Payrovi, “The annihilator graph for modules over commutative rings,” Journal of Algebraic Systems, vol. 9, no. 1, Jun. 2021, doi: 10.22044/jas.2020.9194.1448.
H. M. Salih and A. A. Jund, “Prime ideal graphs of commutative rings,” Indonesian Journal of Combinatorics, vol. 6, no. 1, p. 42, Mar. 2022, doi: 10.19184/ijc.2022.6.1.2.
G. Chartrand, L. Eroh, M. A. Johnson, and O. R. Oellermann, “Resolvability in graphs and the metric dimension of a graph,” Discrete Appl Math (1979), vol. 105, no. 1–3, pp. 99–113, Oct. 2000, doi: 10.1016/S0166-218X(00)00198-0.
F. Okamoto, B. Phinezy, and P. Zhang, “The local metric dimension of a graph,” Mathematica Bohemica, vol. 135, no. 3. pp. 239–255, 2010.
C. Wei, M. F. Nadeem, H. M. Afzal Siddiqui, M. Azeem, J. B. Liu, and A. Khalil, “On Partition Dimension of Some Cycle-Related Graphs,” Math Probl Eng, vol. 2021, 2021, doi: 10.1155/2021/4046909.
K. C. Das, “Atom-bond connectivity index of graphs,” Discrete Appl Math (1979), vol. 158, no. 11, pp. 1181–1188, Jun. 2010, doi: 10.1016/j.dam.2010.03.006.
M. Knor, R. Skrekovski, and A. Tepeh, “Mathematical Aspects of Balaban Index,” Communications in Mathematical and in Computer Chemistry, vol. 79, pp. 685–716, 2018.
M. Faghani and A. R. Ashrafi, “Revised and edge revised Szeged indices of graphs,” Ars Mathematica Contemporanea, vol. 7, no. 1, pp. 153–160, 2014, doi: 10.26493/1855-3974.269.44e.
Copyright (c) 2023 Rian Kurnia, Ahmad Muchlas Abrar, Abdul Gazir Syarifudin, Verrel Rievaldo Wijaya, Nur Ain Supu, Erma Suwastika
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