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

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.

کلیدواژه ها:

Independence number ، Complete graph ، Well-covered ، Clique number ، Vertex decomposable