A study of upper ideal relation graphs of rings

References

• Aalipour, G., Akbari, S., Nikandish, R., Nikmehr, M., Shaveisi, F. (2012). On the coloring of the annihilating-ideal graph of a commutative ring. Discrete Math. 312(17): 2620–2626.
   Web of Science ®Google Scholar
• Akbari, S., Habibi, M., Majidinya, A., Manaviyat, R. (2013). On the idempotent graph of a ring. J. Algebra Appl. 12(6):1350003, 14, 2013.
   Web of Science ®Google Scholar
• Akbari, S., Habibi, M., Majidinya, A., Manaviyat, R. (2015). The inclusion ideal graph of rings. Commun. Algebra 43(6): 2457–246.
   Web of Science ®Google Scholar
• Akbari, S., Maimani, H. R., Yassemi, S. (2003). When a zero-divisor graph is planar or a complete r-partite graph. J. Algebra 270(1): 169–180.
   Web of Science ®Google Scholar
• Anderson, D. F., Badawi, A. (2008). The total graph of a commutative ring. J. Algebra 320(7): 2706–2719.
   Web of Science ®Google Scholar
• Anderson, D. F., Livingston, P. S. (1999). The zero-divisor graph of a commutative ring. J. Algebra 217(2): 434–447.
   Web of Science ®Google Scholar
• Anderson, D. F., Mulay, S. B. (2007). On the diameter and girth of a zero-divisor graph. J. Pure Appl. Algebra 210(2): 543–550.
   Web of Science ®Google Scholar
• Ashrafi, N., Maimani, H. R., Pournaki, M. R., Yassemi, S. (2010). Unit graphs associated with rings. Commun. Algebra 38(8): 2851–2871.
   Web of Science ®Google Scholar
• Atiyah, M. (1994). Introduction to Commutative Algebra. Reading, MA: Addison-Wesley Publishing Company.
   Google Scholar
• Azimi, A., Erfanian, A., Farrokhi, M. D. G. (2013). The Jacobson graph of commutative rings. J. Algebra Appl. 12(3): 1250179, 18.
   Web of Science ®Google Scholar
• Beck, I. (1988). Coloring of commutative rings. J. Algebra 116(1): 208–226.
   Web of Science ®Google Scholar
• Behboodi, M., Rakeei, Z. (2011). The annihilating-ideal graph of commutative rings II. J. Algebra Appl. 10(4): 741–753.
   Web of Science ®Google Scholar
• Chakrabarty, I., Ghosh, S., Mukherjee, T. K., Sen, M. K. (2009). Intersection graphs of ideals of rings. Discrete Math. 309(17): 5381–5392.
   Web of Science ®Google Scholar
• Chartrand, G., Eroh, L., Johnson, M. A., Oellermann, O. R. (2000). Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 105(1–3): 99–113.
   Web of Science ®Google Scholar
• Chattopadhyay, S., Patra, K. L., Sahoo, B. K. (2020). Laplacian eigenvalues of the zero divisor graph of the ring Zn. Linear Algebra Appl. 584: 267–286.
   Web of Science ®Google Scholar
• Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R. (2006). The strong perfect graph theorem. Ann. Math. (2) 164(1): 51–229.
   Web of Science ®Google Scholar
• Chung, F. R. K. (1997). Spectral Graph Theory, volume 92 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI.
   Google Scholar
• Das, A. (2017). On perfectness of intersection graph of ideals of Zn. Discuss. Math. Gen. Algebra Appl. 37(2): 119–126.
   Google Scholar
• Dolžan, D. (2016). The metric dimension of the total graph of a finite commutative ring. Can. Math. Bull. 59(4): 748–759.
   Web of Science ®Google Scholar
• Dolžan, D. (2021). The metric dimension of the annihilating-ideal graph of a finite commutative ring. Bull. Aust. Math. Soc. 103(3): 362–368.
   Web of Science ®Google Scholar
• Ebrahimi, S., Nikandish, R., Tehranian, A., Rasouli, H. (2021). On the strong metric dimension of annihilator graphs of commutative rings. Bull. Malays. Math. Sci. Soc. 44(4): 2507–2517.
   Web of Science ®Google Scholar
• Erlebach, T., Hall, A., Hoffmann, M., Mihaľák, M. (2006). Network discovery and verification with distance queries. In: Algorithms and Complexity, volume 3998 of Lecture Notes in Computer Science. Berlin: Springer, pp. 69–80.
   Google Scholar
• Harary, F., Melter, R. A. (1976). On the metric dimension of a graph. Ars Comb. 2: 191–195.
   Google Scholar
• Lucas, T. G. (2006). The diameter of a zero divisor graph. J. Algebra 301(1): 174–193.
   Web of Science ®Google Scholar
• Ma, X., Feng, M., Wang, K. (2018). The strong metric dimension of the power graph of a finite group. Discrete Appl. Math. 239: 159–164.
   Web of Science ®Google Scholar
• Ma, X., Wong, D. (2016). Automorphism group of an ideal-relation graph over a matrix ring. Linear Multilinear Algebra 64(2): 309–320.
   Web of Science ®Google Scholar
• Magi, P., Jose, S. M., Kishore, A. (2020). Spectrum of the zero-divisor graph on the ring of integers modulo n. J. Math. Comput. Sci. 10(5): 1643–1666.
   Google Scholar
• Maimani, H. R., Salimi, M., Sattari, A., Yassemi, S. (2008). Comaximal graph of commutative rings. J. Algebra 319(4): 1801–1808.
   Web of Science ®Google Scholar
• Mohar, B. (1991). The Laplacian spectrum of graphs. In: Graph Theory, Combinatorics, and Applications. Vol. 2 (Kalamazoo, MI, 1988). New York: Wiley, pp. 871–898.
   Google Scholar
• Nikandish, R., Nikmehr, M. J., Bakhtyiari, M. (2021). Metric and strong metric dimension in cozero-divisor graphs. Mediterr. J. Math. 18(3): 112, 12.
   Web of Science ®Google Scholar
• Nowicki, A. Tables of finite commutative local rings of small orders. https://www-users.mat.umk.pl/∼anow/ps-dvi/tab-05w.pdf.
   Google Scholar
• Ou, S., Wong, D., Tian, F., Zhou, Q. Fixing number and metric dimension of a zero-divisor graph associated with a ring. Linear Multilinear Algebra 69(10): 1789–1802.
   Web of Science ®Google Scholar
• Pirzada, S., Aijaz, M., Imran Bhat, M. (2020). On zero divisor graphs of the rings Zn. Afr. Mat. 31(3–4): 727–737.
   Web of Science ®Google Scholar
• Pirzada, S., Raja, R. (2017). On the metric dimension of a zero-divisor graph. Commun. Algebra 45(4): 1399–1408.
   Web of Science ®Google Scholar
• Pirzada, S., Rather, B. A., Chishti, T. A., Samee, U. (2021). On normalized Laplacian spectrum of zero divisor graphs of commutative ring Zn. Electron. J. Graph Theory Appl. 9(2): 331–345.
   Web of Science ®Google Scholar
• Pirzada, S., Rather, B. A., Ul Shaban, R., Merajuddin, S. (2021). On signless Laplacian spectrum of the zero divisor graphs of the ring Zn. Korean J. Math. 29(1): 13–24.
   Web of Science ®Google Scholar
• Rather, B. A., Pirzada, S., Chishti, T., Alghamdi, A. M. (2022). On normalized laplacian eigenvalues of power graphs associated to finite cyclic groups. Discrete Math. Algorithms Appl. 15: 2250070.
   Web of Science ®Google Scholar
• Schwenk, A. J. (1974). Computing the characteristic polynomial of a graph. In: Graphs and Combinatorics (Proceedings of the Capital Conference on Graph Theory and Combinatorics at the George Washington University, Washington, D.C., 1973). Lecture Notes in Mathematics, Vol. 406. Berlin: Springer, pp. 153–172.
   Google Scholar
• Sebő, A., Tannier, E. (2004). On metric generators of graphs. Math. Oper. Res. 29(2): 383–393.
   Web of Science ®Google Scholar
• Smith, B. (2016). Perfect zero-divisor graphs of Zn. Rose-Hulman Undergrad. Math. J. 17(2): 113–132.
   Google Scholar
• Soleymanivarniab, V., Tehranian, A., Nikandish, R. (2020), “The Metric Dimension of Annihilator Graphs of Commutative Rings. J. Algebra Appl. 19(5): 2050089, 12.
   PubMed Web of Science ®Google Scholar
• Su, H., Li, P. (2014). On the genus of the zero-divisor graph of Zn. Int. J. Comb. 2014: Art. ID 390732, 5.
   Google Scholar
• Wang, H.-J. (2009). Co-maximal graph of non-commutative rings. Linear Algebra Appl. 430(2–3): 633–641.
   Web of Science ®Google Scholar
• West, D. B. (1996). Introduction to Graph Theory, 2nd ed. Upper Saddle River: Prentice Hall.
   Google Scholar
• Ye, M., Wu, T. (2012). Co-maximal ideal graphs of commutative rings. J. Algebra Appl. 11(6): 1250114, 14.
   Web of Science ®Google Scholar
• Young, M. (2015). Adjacency matrices of zero-divisor graphs of integers modulo n. Involve 8(5): 753–761.
   Google Scholar