‎On Power Graph of Some Finite Rings

Document Type : Original Scientific Paper


‎Department of Mathematics, Faculty of Science, ‎University of Qom, ‎Qom‎, ‎I‎. ‎R‎. ‎Iran


‎Consider a ring $R$ with order $p$ or $p^2$‎, ‎and let $\mathcal{P}(R)$ represent its multiplicative power graph‎. ‎For two distinct rings $R_1$ and $R_2$ that possess identity element 1‎, ‎we define a new structure called the unit semi-cartesian product of their multiplicative power graphs‎. ‎This combined structure‎, ‎denoted as $G.H$‎, ‎is constructed by taking the Cartesian product of the vertex sets $V(G) \times V(H)$‎, ‎where $G = \mathcal{P}(R1)$ and $H = \mathcal{P}(R2)$‎. ‎The edges in $G.H$ are formed based on specific conditions‎: ‎for vertices $(g,h)$ and $(g^\prime,h^\prime)$‎, ‎an edge exists between them if $g = g^\prime$‎, ‎$g$ is a vertex in $G$‎, ‎and the product $hh^\prime$ forms a vertex in $H$‎.
‎Our exploration focuses on understanding the characteristics of the multiplicative power graph resulting from the unit semi-cartesian product $\mathcal{P}(R1).\mathcal{P}(R2)$‎, ‎where $R_1$ and $R_2$ represent distinct rings‎. ‎Additionally‎, ‎we offer insights into the properties of the multiplicative power graphs inherent in rings of order $p$ or $p^2$‎.


Main Subjects

[1] M. Soleimani, M. H. Naderi and A. R. Ashrafi, Tensor product of the power
graphs of some finite rings, Facta Univ. Ser. Math. Inform. 34 (1) (2019)
101 − 122, https://doi.org/10.22190/FUMI1901101S.
[2] F. Mahmudi and M. Soleimani, Some results on Maximal Graph of a Commutative Ring, 47th Annual Iranian Mathematics Conference, Tehran, Iran,
[3] F. Mahmudi, M. Soleimani and M. H. Naderi, Some properties of the maximal
graph of a commutative ring, Southeast Asian Bull. Math. 43 (4) (2019)
525 − 536.
[4] M. Soleimani, F. Mahmudi and M. H. Naderi, Some results on the maximal
graph of commutative rings, Adv. Studies: Euro-Tbilisi Math. J. 16 (1) (2023)
21 − 26, https://doi.org/10.32513/asetmj/1932200823104.
[5] P. J. Cameron, The power graph of a finite group, II, J. Group Theory, 13
(6) (2010) 779 − 783, https://doi.org/10.1515/jgt.2010.023.
[6] A. V. Kelarev and S. J. Quinn, A combinatorial property and power graphs
of groups, Contributions to general algebra, 12 (1999), 229 − 235, Heyn, Klagenfurt, 2000.
[7] A. V. Kelarev and S. J. Quinn, A combinatorial property and power graphs
of semigroups, Comment. Math. Univ. Carolin. 45 (1) (2004) 1 − 7.
[8] A. V. Kelarev and S. J. Quinn, Directed graphs and combinatorial properties of semigroups, J. Algebra, 251 (1) (2002) 16 − 26,
[9] A. V. Kelarev, S. J. Quinn and R. Smol´ikov´a, Power graphs and semigroups of matrices, Bull. Aust. Math. Soc. 63 (2) (2001) 341 − 344,
[10] B. Fine, Classification of finite rings of order p
, Math. Mag. 66 (1993) 248 −
252, https://doi.org/10.1080/0025570X.1993.11996133.
[11] A. Hamzeh, Some graph polynomials of the power graph
and its supergraphs, Math. Interdisc. Res. 5 (2020) 13 − 22,
[12] A. Hamzeh and A. R. Ashrai, Some remarks on the order supergraph of the
power graph of a finite group, Int. Electron. J. Algebra, 26 (2019) 1 − 12,
[13] A. Hamzeh and A. R. Ashrai, The order supergraph of the power
graph of a finite group, Turkish J. Math. 42 (2018) 1978 − 1989, 4.
[14] A. Hamzeh and A. R. Ashrai, Automorphism groups of supergraphs of
the power graph of a finite group, Eur. J. Comb. 60 (2017) 82 − 88,
[15] A. Hamzeh and A. R. Ashrai, Spectrum and L-spectrum of the power graph
and its main supergraph for certain finite groups, Filomat, 31 (16) (2017)
5323 − 5334, https://doi.org/10.2298/FIL1716323H.
[16] J. Abawajy, A. Kelarev, M. Chowdhury, Power graphs: a survey, Electron. J. Graph Theory Appl. 1 (2) (2013) 125 − 147,
[17] B. Bollobás, Graph Theory, An Introductory Course, Springer, New York,
[18] R. Diestel, Graph Theory, Springer, New York, 1997.
[19] A. V. Kelarev, Ring Constructions and Applications, World Scientific, 2002.
[20] A. V. Kelarev, Graph Algebras and Automata, Marcel Dekker, New York,