Hamiltonian property of intersection graph of zero divisors of the ring extbf{$Z_n$}

Hamiltonian property of intersection graph of zero divisors of the ring extbf{$Z_n$}

**Authors : **

Shaik Sajana^{1*} and D Bharathi^{2}

**Author Address : **

^{1,2}Department of Mathematics, S.V. University, Tirupati-517502, India.

*Corresponding author.

**Abstract : **

The intersection graph $G_{Z}^{’}(Z_{n})$ of zero-divisors of the ring $Z_n$, the ring of integers modulo $n$ is a simple undirected graph with the vertex set is $Z(Z_{n})^{*} = Z(Z_{n})setminus lbrace0 brace$, the set of all nonzero zero-divisors of the ring $Z_n$ and for any two distinct vertices are adjacent if and only if their corresponding principal ideals have a nonzero intersection. We determine some results concerning the necessary and sufficient condition for the graph $G_{Z}^{’}(Z_{n})$ is Hamiltonian. Also, we investigate for all values of for which the graph $G_{Z}^{’}(Z_{n})$ is Hamiltonian and as an example we show that how the results give as easy proof of the existence of a Hamilton cycle.

**Keywords : **

Finite commutative ring, Zero-divisors, Principal ideals, Intersection graph, Hamilton Cycle.

**DOI : **

**Article Info : **

*Received : * September 24, 2017; *Accepted : * November 30, 2017.