Wiener index of graphs over rings: a survey

References

• Ahmadi, M. R, Nezhad, R. J. (2011). Energy and Wiener index of zero-divisor graphs. Iran. J. Math. Chem. 2(1): 45–51.
   Google Scholar
• Afkhami, M., Barati, Z, Khashyarmanesh, K. (2014). When the unit, unitary and total graphs are ring graphs and outer planar. Rocky Mountain J. Math. 44(3): 705–716.
   Web of Science ®Google Scholar
• Koam, A. N. A., Ahmad, A, Haider, A. (2019). On eccentric topological indices based on edges of zero-divisor graphs. Symmetry 11(7): 907.
   Google Scholar
• Anderson, D. F., Asir, T., Tamizh Chelvam, T, Badawi, A. (2021). Graphs from Rings. New York: Springer International Publishing.
   Google Scholar
• Anderson, D. F, Badawi, A. (2008). The total graph of 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
• Ashrafi, N., Maimani, H. R., Pournaki, M. R, Yassemi, S. (2010). Unit graphs associated with rings. Comm. Algebra 38(8): 2851–2871.
   Web of Science ®Google Scholar
• Asir, T, Rabikka, V. (2022). The Wiener index of the zero-divisor graph of Zn. Discrete Appl. Math. 319: 461–471.
   Web of Science ®Google Scholar
• Asir, T., Rabikka, V, Su, H. (2022). On Wiener index of unit graph associated with a commutative ring. Algebra Colloq. 29(02): 221–230.
   Web of Science ®Google Scholar
• Asir, T, Tamizh Chelvam, T. (2014). On the genus of generalized unit and unitary Cayley graphs of a commutative ring. Acta Math. Hung. 142(2): 444–458.
   Web of Science ®Google Scholar
• Beck, I. (1988). Coloring of commutative rings. J. Algebra 116(1): 208–226.
   Web of Science ®Google Scholar
• Bhavanari, S., Kuncham, S. P, Dasari, N. (2010). Prime graph of a ring. J. Comb. Inf. Syst. Sci 35(1–2): 27–42.
   Google Scholar
• Chartrand, G. (2000). Lesniak, Graphs and Digraphs. New York: Chapman & Hall/CRC.
   Google Scholar
• Dobrynin, A., Entringer, R, Gutman, I. (2001). Wiener index of trees: theory and applications. Acta Appl. Math. 66(3): 211–249.
   Web of Science ®Google Scholar
• Dummit, D. S, Foote, R. M. (2003). Abstract Algebra. Hoboken, New Jersey: Wiley.
   Google Scholar
• Grimaldi, R. P. (1990). Graphs from rings. Proceedings of the 20th Southeastern Conference on Combinatorics. Graph Theory and Computing (Boca Raton, FL), Congr. Numer 71: 95–103.
   Google Scholar
• Gutman, I, Polansky, O. E. (1986). Mathematical Concepts in Organic Chemistry, Springer-Verlag: New York, NY, USA.
   Google Scholar
• Hosoya, H. (1971). Topological index. a newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. BCSJ. 44(9): 2332–2339.
   Google Scholar
• Joshi, S. S, Pawar, K. F. (2018). Energy, Wiener index and Line graph of prime graph of a ring. Int. J. Math. Combin. 3: 74–80.
   Google Scholar
• Klein, D. J., Lukovits, I, Gutman, I. (1995). On the definition of the hyper-Wiener index for cycle-containing structures. J. Chem. Inf. Comput. Sci. 35(1): 50–52.
   Google Scholar
• Knor, M., Lužar, B., Škrekovski, R, Gutman, I. (2014). On Wiener index of common neighborhood graphs. Commun. Math. Comput. Chem. 72: 321–332.
   Web of Science ®Google Scholar
• Maimani, H. R., Pournaki, M. R., Tehranian, A, Yassemi, S. (2011). Graphs attached to rings revisited. Arab. J. Sci. Eng. 36(6): 997–1011.
   Web of Science ®Google Scholar
• Maimani, H. R., Pournaki, M. R, Yassemi, S. (2011). Necessary and sufficient conditions for unit graphs to be Hamiltonian. Pacific J. Math. 249(2): 419–429.
   Web of Science ®Google Scholar
• Maimani, H. R., Pournaki, M. R, Yassemi, S. (2010). Weakly perfect graphs arising from rings. Glasgow Math. J. 52(3): 417–425.
   Google Scholar
• Maimani, H. R., Pournaki, M. R, Yassemi, S. (2010). Rings which are generated by their units: a graph theoretical approach. Elem. Math. 65(1): 17–25.
   Google Scholar
• Maimani, H. R., Pournaki, M. R, Yassemi, S. (2010). A class of weakly perfect graphs. Czech. Math. J. 60(4): 1037–1041.
   Web of Science ®Google Scholar
• Maimani, H. R., Wickham, C, Yassemi, S. (2012). Rings whose total graphs have genus at most one. Rocky Mountain J. Math. 42(5): 1551–1560.
   Web of Science ®Google Scholar
• Nikmehr, M. J., Heidarzadeh, L, Soleimani, N. (2014). Calculating different topological indices of total graph of Zn. Stud. Sci. Math. Hung. 51(1): 133–140.
   Web of Science ®Google Scholar
• Pirzada, S., Aijaz, M, Imran Bhat, M. (2020). On zero-divisor graphs of the rings Zn. Afr. Math. 31: 727–737.
   Web of Science ®Google Scholar
• Pranjali, M. Acharya, (2015). Energy and Wiener index of unit graph. Appl. Math. Inf. Sci. 9(3): 1339–1343.
   Google Scholar
• Randic, M. (1993). Novel molecular descriptor for structure-property studies. Chem. Phys. Lett. 211(4–5): 478–483.
   Web of Science ®Google Scholar
• Reddy, B. S., Jain, R. S, Laxmikanth, N. (2017). Eigenvalues and Wiener index of the zero-divisor graph Γ(Zn). arXiv:1707.05083 [Math.RA].
   Google Scholar
• Selvakumar, K., Gangaeswari, P, Arunkumar, G. (2022). The Wiener index of the Zero-divisor graph of a finite commutative ring with unity. Discrete Appl. Math. 311: 72–84.
   Web of Science ®Google Scholar
• Sheela, S, Prakash, O. (2018). Adjacency matrix and energy of the line graph of Γ(Zn). arXiv:1806.08944v1 [Math.CO].
   Google Scholar
• Sheela, S, Prakash, O. (2017). Energy and Wiener index of total graph over ring Zn. Electron. Notes Discrete Math. 63: 485–495.
   Google Scholar
• Tamizh Chelvam, T, Asir, T. (2013). On the genus of the total graph of a commutative ring. Comm. Algebra 41(1): 142–153.
   Web of Science ®Google Scholar
• Tamizh Chelvam, T, Asir, T. (2013). On the total graph and its complement of a commutative ring. Comm. Algebra 41(10): 3820–3835.
   Web of Science ®Google Scholar
• Wiener, H. (1947). Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69(1): 17–20.
   PubMed Web of Science ®Google Scholar