A REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF A FINITE RING

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 53, Issue 4, 2016, pp.1197-1211
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.b150601

Title & Authors

A REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF A FINITE RING

Naghipour, Ali Reza; Rezagholibeigi, Meysam;

Naghipour, Ali Reza; Rezagholibeigi, Meysam;

Abstract

Let R be a finite commutative ring with nonzero identity. We define to be the graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exists a unit element u of R such that x + uy is a unit of R. This graph provides a refinement of the unit and unitary Cayley graphs. In this paper, basic properties of are obtained and the vertex connectivity and the edge connectivity of are given. Finally, by a constructive way, we determine when the graph is Hamiltonian. As a consequence, we show that has a perfect matching if and only if is an even number.

Keywords

Cayley graphs;Clique number;Hamiltonian graphs;finite rings;matchings;unit graphs;

Language

English

References

1.

R. Akhtar, M. Boggess, T. Jackson-Henderson, I. Jimenez, R. Karpman, A. Kinzel, and D. Pritikin, On the unitary Cayley graph of a finite ring, Electron. J. Combin. 16 (2009), no. 1, Research Paper 117, 13 pp.

2.

N. Ashrafi, H. R. Maimani, M. R. Pournaki, and S. Yassemi, Unit graphs associated with rings, Comm. Algebra 38 (2010), no. 8, 2851-2871.

3.

T. Asir and T. T. Chelvam, On the genus of generalized unit and unitary Cayley graphs of a commutative ring, Acta Math. Hungar. 142 (2014), no. 2, 444-458.

4.

M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company, 1969.

5.

J. A. Bondy and U. S. R. Murty, Graph Theory, Graduate Texts in Mathematics, 244 Springer, New York, 2008.

6.

H. Chen, Related to Stable Conditions, World Scientic (Series in Algebra 11), Hakensack, NJ, 2011.

7.

A. K. Das, H. R. Maimani, M. R. Pournaky, and S. Yassemi, Nonplanarity of unit graphs and classification of the toroidal ones, Pacific J. Math. 268 (2014), no. 2, 371-387.

8.

E. Fuchs, Longest induced cycles in circulant graphs, Electron. J. Combin. 14 (2005), no. 1, Research Paper 52, 12 pp.

9.

K. Khashyarmanesh and M. R. Khorsandi, A generalization of the unit and unitary Cayley graphs of a commutative ring, Acta Math. Hungar. 137 (2012), no. 4, 242-253.

10.

D. Kiani and M. M. H. Aghaei, On the unitary Cayley graphs of a ring, Electron. J. Combin. 19 (2012), no. 2, Research paper 10, 10 pp.

11.

D. Kiani, M. M. H. Aghaei, Y. Meemark, and B. Suntornpoch, Energy of unitary Cayley graphs and gcd-graphs, Linear Algebra Appl. 435 (2011), no. 6, 1336-1343.

12.

W. Klotz and T. Sander, Some properties of unitary Cayley graphs, Electron. J. Combin. 14 (2007), no. 1, Research Paper 45, 12 pp.

13.

T. Y. Lam, Bass's work in ring theory and projective modules, Algebra K-theory, groups, and education (New York, 1997), 83-124, Contemp. Math., 243, Amer. Math. Soc., Providence, RI, 1999.

14.

T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, Inc. 2001.

15.

X. Liu and S. Zhou, Spectral properties of unitary Cayley graphs of finite commutative rings, Electron. J. Combin. 19 (2012), no. 4, Research paper 13, 19 pp.

16.

H. R. Maimani, M. R. Pournaki, A. Tehranian, and S. Yassemi, Graphs attached to rings revisited, Arab. J. Sci. Eng. 36 (2011), no. 6, 997-1011.

17.

H. R. Maimani, M. R. Pournaki, and S. Yassemi, Necessary and sucient conditions for unit graphs to be Hamiltonian, Pacific J. Math. 249 (2011), no. 2, 419-429.

18.

P. K. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176 (1995), no. 1, 124-127.

19.

H. Su and Y. Zhou, On the girth of the unit graph of a ring, J. Algebra Appl. 13 (2014), no. 2, 1350082, 12 pp.

20.

D. B. West, Introduction to Graph Theory, second ed., Prentice-Hall, 2000.