A class of well-covered and vertex decomposable graphs arising from rings

Document Type : Research Paper


1 Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU), Tehran, Iran.

2 Department of Mathematics, Jundi-Shapur University of Technology, Dezful, Iran.


Let $ \mathbb {Z}_{n} $ be the ring of integers modulo $ n $. The unitary Cayley graph of $ \mathbb {Z}_{n} $ is defined as the graph $ G( \mathbb {Z}_{n} ) $ with the vertex set $ \mathbb {Z}_{n} $ and two distinct vertices $a,b$ are adjacent if and only if  $a-b\in U\left( \mathbb {Z}_{n}\right)$, where $ U\left( \mathbb {Z}_{n}\right) $ is the set of units of $ \mathbb {Z}_{n} $. Let $\Gamma ( \mathbb {Z}_{n} ) $ be the complement of $ G( \mathbb {Z}_{n} )  $. In this paper, we determine the independence number of $ \Gamma ( \mathbb {Z}_{n} ) $. Also it is proved that $\ \Gamma ( \mathbb {Z}_{n} ) $ is well-covered.  Among other things, we provide condition under which $ \Gamma ( \mathbb {Z}_{n} ) $ is vertex decomposable.


[1] A. Alibemani, M. Bakhtyiari, R. Nikandish, M.J. Nikmehr, The annihilator ideal graph of a commutative ring, J. Korean Math. Soc. 52 (2015), 417-429.
[2] R. Akhtar, M. Boggess, T. Jackson-Henderson, I. Jimenez, R. Karpman, A. Kinzel, D. Pritikin, On the unitary Cayley graph of a nite ring, Electron. J. Combin. 16 (2009) R117.
[3] D.F. Anderson, P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999),434-447.
[4] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison Wesley Publishing Co, Reading, Mate.-London-Don Mills, Ont, 1969.
[5] A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra, 42 (2014), 108-121.
[6] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), 208-226.
[7] M. Behboodi, Z. Rakeie, The annihilating-ideal graph of a commutative ring I, J. Algebra Appl. 10 (2011), 727-739.
[8] M. Behboodi, Z. Rakeie, The annihilating-ideal graph of a commutative ring II, J. Algebra Appl. 10 (2011), 741-753.
[9] R. Belsho and J. Chapman, Planar zero-divisor graphs, J. Algebra 316 (2007), 471-480.
[10] A. Bjorner, M. L. Wachs, Shellable nonpure complexes and posets. I. Trans. Amer. Math. Soc. 348 (1996), 1299-1327.
[11] A. Bjorner, M. L. Wachs, Shellable nonpure complexes and posets. II. Trans. Amer. Math. Soc. 349 (1997), 3945-3975.
[12] N. Jahanbakhsh Basharlou, M. J. Nikmehr, R. Nikandish On generalized zero-divisor graph associated with a commutative ring. Italian J. Pure Appl. Math., 39 (2018), 128-139.
[13] S. Kiani, H. R. Maimani, S. Yassemi, Well-covered and cohen macaulay unitary cayley graphs. Acta Math. Hungarica 144 (2014), 92-98.
[14] M.Mahmoudi, A. Mousivand, M. Crupi, G. Rinaldo, N. Terai, S. Yassemi, Vertex Decomposibility and Regularityof Very Well-Covered Graphs. J. Pure  Appl. Alg., 215 (2011), 2473-2480.
[15] A. Mousivand, S. A. S. Fakhari, S. Yassemi, A new construction for Cohen-Macaulay graphs. Comm Algebra. 43 (2015), 5104-5112.
[16] J. S. Provan, L. J. Billera, Decompositions of simplicial complexes related to diameters of convex polyhedra. Math. Oper. Res. 5 (1980), 576-594.
[17] T. Tamizh Chelvam, K. Selvakumar, On the connectivity of the annihilating-ideal graphs, Discuss. Math. Gen. Alg. Appl., 35 (2015), 195-204.
[18] R. H. Villarreal, Cohen-Macaulay graphs. Manuscripta Math. 66 (1990), 277-293.
[19] R. Woodroofe, Vertex decomposable graphs and obstructions to shellability. Proc. Amer. Math. Soc. 137 (2009) 3235-3246.