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.


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