Unit and unitary Cayley graphs for the ring of Eisenstein integers modulo \(n\). (English) Zbl 1486.05130

Jahani-Nezhad, Reza; Bahrami, Ali

Ural Math. J. 7, No. 2, 43-50 (2021).

Summary: Let \({E}_n\) be the ring of Eisenstein integers modulo \(n\). We denote by \(G({E}_n)\) and \(G_{{E}_n}\), the unit graph and the unitary Cayley graph of \({E}_n \), respectively. In this paper, we obtain the value of the diameter, the girth, the clique number and the chromatic number of these graphs. We also prove that for each \(n>1\), the graphs \(G(E_n)\) and \(G_{E_n}\) are Hamiltonian.

MSC:

05C25	Graphs and abstract algebra (groups, rings, fields, etc.)
05C45	Eulerian and Hamiltonian graphs

Keywords:

unit graph; unitary Cayley graph; Eisenstein integers; Hamiltonian graph

References:

[1]	Aalipour G., Akbari S., “On the Cayley graph of a commutative ring with respect to its zero-divisors”, Comm. Algebra, 44:4 (2016), 1443-1459 · Zbl 1341.05114
[2]	Akbari S., Estaji E., Khorsandi M. R., “On the unit graph of a non-commutative ring”, Algebra Colloq., 22, 817-822 · Zbl 1328.05075
[3]	Akhtar R., Jackson-Henderson T., Karpman R., Boggess M., Jiménez I., Kinzel A., Pritikin D., “On the unitary Cayley graph of a finite ring”, Electron. J. Combin., 16:1 (2009), R117 · Zbl 1230.05149
[4]	Alkam O., Abu Osba E., “On Eisenstein integers modulo \(n\)”, Int. Math. Forum., 5:22 (2010), 1075-1082 · Zbl 1222.11121
[5]	Anderson D. F., Badawi A., “The total graph of a commutative ring”, J. Algebra, 320:7 (2008), 2706-2719 · Zbl 1158.13001
[6]	Anderson D. F., Livingston P. S., “The zero-divisor graph of a commutative ring”, J. Algebra, 217:2 (1999), 434-447 · Zbl 0941.05062
[7]	Ashrafi N., Maimani H. R., Pournaki M. R., Yassemi S., “Unit graphs associated with rings.”, Comm. Algebra, 38 (2010), 2851-2871 · Zbl 1219.05150
[8]	Atiyah M. F., MacDonald I. G., Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Menlo Park, California, London, Don Mills, Ontario, 1969, 128 pp. · Zbl 0175.03601
[9]	Bahrami A., Jahani-Nezhad R., “Unit and unitary Cayley graphs for the ring of Gaussian integers modulo \(n\)”, Quasigroups Related Systems, 25:2 (2017), 189-200 · Zbl 1401.13024
[10]	Beck I., “Coloring of commutative rings”, J. Algebra, 116:1 (1988), 208-226 · Zbl 0654.13001
[11]	Bondy J. A., Murty U. S. R., Graph Theory with Applications, North-Holland, New York, Amsterdam, Oxford, 1976, 264 pp. · Zbl 1226.05083
[12]	Cayley A., “Desiderata and Suggestions: No. 2. The Theory of Groups: Graphical Representation”, Amer. J. Math., 1:2 (1878), 174-176 · JFM 10.0105.02
[13]	Chung F. R. K., “Diameters and eigenvalues”, J. Amer. Math. Soc., 2:2 (1989), 187-196 · Zbl 0678.05037
[14]	Dejter I. J., Giudici R. E, “On unitary Cayley graphs”, J. Combin. Math. Comput., 18 (1995), 121-124 · Zbl 0832.05052
[15]	Diestel R., Graph Theory, Springer-Verlag, Berlin, Heidelberg, 2001, 428 pp. · Zbl 1375.05002
[16]	Grimaldi R. P., “Graphs from rings”, Congr. Numer., 17 (1990), 95-103 · Zbl 0747.05091
[17]	Ireland K., Rosen M., A Classical Introduction to Modern Number Theory, Springer-Verlag, NY, 1990, 394 pp. · Zbl 0712.11001
[18]	Khashyarmanesh K., Khorsandi M. R., “A generalization of the unit and unitary Cayley graphs of a commutative ring”, Acta Math. Hungar., 137 (2012), 242-253 · Zbl 1289.05205
[19]	Kiani D., Aghaei M. M. H., “On the unitary Cayley graph of a ring”, Electron. J. Combin., 19:2 (2012), P10 · Zbl 1264.05066
[20]	Lanski C., Maróti A., “Ring elements as sums of units”, Cent. Eur. J. Math., 7 (2009), 395-399 · Zbl 1185.16026
[21]	Maimani H. R., Pournaki M. R., Yassemi S., “Weakly perfect graphs arising from rings”, Glasg. Math. J., 52:3 (2010), 417-425 · Zbl 1243.05085
[22]	Maimani H. R., Pournaki M. R., Yassemi S., “Necessary and sufficient conditions for unit graphs to be Hamiltonian”, Pacific J. Math., 249:2 (2011), 419-429 · Zbl 1214.05074