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

Document Type : Research Paper

Authors

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.

10.29252/as.2020.1795

Abstract

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.

Keywords

#### References

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