Jacobson-Type Graphs Over Rings: Induced Subgraphs and Isomorphisms
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Keywords:
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Jacobson Graph, Jacobson Radical, Induced Subgraph, Graph Isomorphism, Matrix Rings, Finite Commutative Rings
Abstract
Jacobson-type graphs provide a combinatorial framework for studying structural properties of finite commutative rings. We address two foundational questions: the behavior of these graphs under subring inclusions, and conditions under which distinct Jacobson-type constructions are isomorphic. This paper proves that whenever S is a subring of a finite commutative ring R, the Jacobson-type graphs of S arise naturally as induced subgraphs of the corresponding graphs of R, with finiteness playing an essential role in sustaining this embedding. We further characterize precisely when the Jacobson graph, the nn-array Jacobson graph, and the matrix Jacobson graph over a ring are isomorphic. These results establish a correspondence between algebraic relations of rings and combinatorial relations of their associated graphs.
References
Akbari, S., Khojasteh, S., & Yousefzadehfard, A. (2015). The proof of a conjecture in Jacobson graph of a commutative ring. Journal of Algebra and Its Applications, 14(10), 1550107. doi:10.1142/S0219498815501078
Akbari, S., Maimani, H. R., & Yassemi, S. (2003). When a zero-divisor graph is planar or a complete r-partite graph. Journal of Algebra, 270(1), 169–180. doi:10.1016/S0021-8693(03)00370-3
Akbari, S., & Mohammadian, A. (2004). On the zero-divisor graph of a commutative ring. Journal of Algebra, 274(2), 847–855. doi:10.1016/S0021-8693(03)00435-6
Anderson, D. F., Axtell, M. C., & Stickles, J. A. (2011). Zero-divisor graphs in commutative rings. In M. Fontana, S.-E. Kabbaj, B. Olberding, & I. Swanson (Eds.), Commutative algebra: Noetherian and non-Noetherian perspectives (pp. 23–45). New York, NY: Springer. doi:10.1007/978-1-4419-6990-3_2
Anderson, D. F., & Livingston, P. S. (1999). The zero-divisor graph of a commutative ring. Journal of Algebra, 217(2), 434–447. doi:10.1006/jabr.1998.7840
Anderson, D. D., & Naseer, M. (1993). Beck's coloring of a commutative ring. Journal of Algebra, 159(2), 500–514. doi:10.1006/jabr.1993.1171
Atiyah, M. F., & Macdonald, I. G. (1969). Introduction to commutative algebra. Reading, MA: Addison-Wesley.
Azimi, A., Erfanian, A., & Farrokhi D. G., M. (2013). The Jacobson graph of commutative rings. Journal of Algebra and Its Applications, 12(3), 1250179. doi:10.1142/S0219498812501794
Azimi, A., Erfanian, A., & Farrokhi D. G., M. (2014). Isomorphisms between Jacobson graphs. Rendiconti del Circolo Matematico di Palermo Series 2, 63(2), 277–286. doi:10.1007/s12215-014-0155-5
Azimi, A., & Farrokhi D. G., M. (2017). Cycles and paths in Jacobson graphs. Ars Combinatoria, 134, 61–74.
Beck, I. (1988). Coloring of commutative rings. Journal of Algebra, 116(1), 208–226. doi:10.1016/0021-8693(88)90202-5
Diestel, R. (2017). Graph theory (5th ed.). Berlin, Germany: Springer-Verlag.
Fattahi, Z., Erfanian, A., & Alinejad, M. (2019). Some new results on Jacobson graphs. Rendiconti del Circolo Matematico di Palermo Series 2, 68(1), 129–137. doi:10.1007/s12215-018-0340-x
Ghayour, H., Erfanian, A., & Azimi, A. (2018). Some results on the Jacobson graph of a commutative ring. Rendiconti del Circolo Matematico di Palermo Series 2, 67(1), 33–41. doi:10.1007/s12215-017-0299-5
Ghayour, H., Erfanian, A., Azimi, A., & Farrokhi D. G., M. (2017). n-array Jacobson graphs. Bulletin of the Iranian Mathematical Society, 43(7), 2137–2152.
Herstein, I. N. (1991). Topics in algebra (2nd ed.). Hoboken, NJ: John Wiley & Sons.
Humaira, S., Astuti, P., Muchtadi-Alamsyah, I., & Baskoro, E. T. (2023). Jacobson graph of matrix rings. Mathematics and Statistics, 11(5), 794–801. doi:10.13189/ms.2023.110505
Humaira, S., Astuti, P., Muchtadi-Alamsyah, I., & Erfanian, A. (2022). The matrix Jacobson graph of finite commutative rings. Electronic Journal of Graph Theory and Applications, 10(1), 181–198. doi:10.5614/ejgta.2022.10.1.12
Humaira, S., Astuti, P., Muchtadi-Alamsyah, I., & Erfanian, A. (2020). The matrix Jacobson graph of fields. Journal of Physics: Conference Series, 1538(1), 012008. doi:10.1088/1742-6596/1538/1/012008
Matsumura, H. (1986). Commutative ring theory (Cambridge Studies in Advanced Mathematics, Vol. 8). Cambridge, UK: Cambridge University Press.
McDonald, B. R. (1974). Finite rings with identity. New York, NY: Marcel Dekker.
Ore, O. (1962). Theory of graphs (Colloquium Publications, Vol. 38). Providence, RI: American Mathematical Society.
Parsapour, A., Khashyarmanesh, K., Afkhami, M., & Ahmad Javaheri, K. (2015). Classification of finite commutative rings with planar, toroidal, and projective line graphs associated with Jacobson graphs. Mathematical Notes, 98(5), 813–819. doi:10.1134/S0001434615110103
Scheinerman, E. R., & Ullman, D. H. (1997). Fractional graph theory: A rational approach to the theory of graphs. New York, NY: Wiley-Interscience.
Sharma, P. K., & Bhatwadekar, S. M. (1995). A note on graphical representation of rings. Journal of Algebra, 176(1), 124–127. doi:10.1006/jabr.1995.1236
Akbari, S., Maimani, H. R., & Yassemi, S. (2003). When a zero-divisor graph is planar or a complete r-partite graph. Journal of Algebra, 270(1), 169–180. doi:10.1016/S0021-8693(03)00370-3
Akbari, S., & Mohammadian, A. (2004). On the zero-divisor graph of a commutative ring. Journal of Algebra, 274(2), 847–855. doi:10.1016/S0021-8693(03)00435-6
Anderson, D. F., Axtell, M. C., & Stickles, J. A. (2011). Zero-divisor graphs in commutative rings. In M. Fontana, S.-E. Kabbaj, B. Olberding, & I. Swanson (Eds.), Commutative algebra: Noetherian and non-Noetherian perspectives (pp. 23–45). New York, NY: Springer. doi:10.1007/978-1-4419-6990-3_2
Anderson, D. F., & Livingston, P. S. (1999). The zero-divisor graph of a commutative ring. Journal of Algebra, 217(2), 434–447. doi:10.1006/jabr.1998.7840
Anderson, D. D., & Naseer, M. (1993). Beck's coloring of a commutative ring. Journal of Algebra, 159(2), 500–514. doi:10.1006/jabr.1993.1171
Atiyah, M. F., & Macdonald, I. G. (1969). Introduction to commutative algebra. Reading, MA: Addison-Wesley.
Azimi, A., Erfanian, A., & Farrokhi D. G., M. (2013). The Jacobson graph of commutative rings. Journal of Algebra and Its Applications, 12(3), 1250179. doi:10.1142/S0219498812501794
Azimi, A., Erfanian, A., & Farrokhi D. G., M. (2014). Isomorphisms between Jacobson graphs. Rendiconti del Circolo Matematico di Palermo Series 2, 63(2), 277–286. doi:10.1007/s12215-014-0155-5
Azimi, A., & Farrokhi D. G., M. (2017). Cycles and paths in Jacobson graphs. Ars Combinatoria, 134, 61–74.
Beck, I. (1988). Coloring of commutative rings. Journal of Algebra, 116(1), 208–226. doi:10.1016/0021-8693(88)90202-5
Diestel, R. (2017). Graph theory (5th ed.). Berlin, Germany: Springer-Verlag.
Fattahi, Z., Erfanian, A., & Alinejad, M. (2019). Some new results on Jacobson graphs. Rendiconti del Circolo Matematico di Palermo Series 2, 68(1), 129–137. doi:10.1007/s12215-018-0340-x
Ghayour, H., Erfanian, A., & Azimi, A. (2018). Some results on the Jacobson graph of a commutative ring. Rendiconti del Circolo Matematico di Palermo Series 2, 67(1), 33–41. doi:10.1007/s12215-017-0299-5
Ghayour, H., Erfanian, A., Azimi, A., & Farrokhi D. G., M. (2017). n-array Jacobson graphs. Bulletin of the Iranian Mathematical Society, 43(7), 2137–2152.
Herstein, I. N. (1991). Topics in algebra (2nd ed.). Hoboken, NJ: John Wiley & Sons.
Humaira, S., Astuti, P., Muchtadi-Alamsyah, I., & Baskoro, E. T. (2023). Jacobson graph of matrix rings. Mathematics and Statistics, 11(5), 794–801. doi:10.13189/ms.2023.110505
Humaira, S., Astuti, P., Muchtadi-Alamsyah, I., & Erfanian, A. (2022). The matrix Jacobson graph of finite commutative rings. Electronic Journal of Graph Theory and Applications, 10(1), 181–198. doi:10.5614/ejgta.2022.10.1.12
Humaira, S., Astuti, P., Muchtadi-Alamsyah, I., & Erfanian, A. (2020). The matrix Jacobson graph of fields. Journal of Physics: Conference Series, 1538(1), 012008. doi:10.1088/1742-6596/1538/1/012008
Matsumura, H. (1986). Commutative ring theory (Cambridge Studies in Advanced Mathematics, Vol. 8). Cambridge, UK: Cambridge University Press.
McDonald, B. R. (1974). Finite rings with identity. New York, NY: Marcel Dekker.
Ore, O. (1962). Theory of graphs (Colloquium Publications, Vol. 38). Providence, RI: American Mathematical Society.
Parsapour, A., Khashyarmanesh, K., Afkhami, M., & Ahmad Javaheri, K. (2015). Classification of finite commutative rings with planar, toroidal, and projective line graphs associated with Jacobson graphs. Mathematical Notes, 98(5), 813–819. doi:10.1134/S0001434615110103
Scheinerman, E. R., & Ullman, D. H. (1997). Fractional graph theory: A rational approach to the theory of graphs. New York, NY: Wiley-Interscience.
Sharma, P. K., & Bhatwadekar, S. M. (1995). A note on graphical representation of rings. Journal of Algebra, 176(1), 124–127. doi:10.1006/jabr.1995.1236
Authors
Humaira, S., Purwasih, I. A. ., Awanis, Z. Y. ., Sofyan, D. K. ., Astuti, P. ., & Muchtadi Alamsyah, I. (2026). Jacobson-Type Graphs Over Rings: Induced Subgraphs and Isomorphisms. Science and Technology Indonesia, 11(4), 1715–1723. https://doi.org/10.26554/sti.2026.11.4.1715-1723
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