Home > RHUMJ > Vol. 17 (2016) > Iss. 2

#### Article Title

#### Abstract

The zero-divisor graph of a ring *R*, denoted &Gamma(*R*), is the graph whose vertex set is the collection of zero-divisors in *R*, with edges between two distinct vertices *u* and *v* if and only if *uv*=0. In this paper, we restrict our attention to &Gamma(**Z**_{n}), the zero-divisor graph of the ring of integers modulo *n*. Specifically, we determine all values of *n* for which &Gamma(**Z**_{n}) is perfect. Our classification depends on the prime factorization of *n*, with relatively simple prime factorizations corresponding to perfect graphs. In fact, for values of *n* with at most two distinct prime factors, &Gamma(**Z**_{n}) is perfect; for *n* with at least 5 distinct prime factors, &Gamma(**Z**_{n}) is not perfect; and for *n* with either three or four distinct prime factors, &Gamma(**Z**_{n}) is perfect only in the cases where *n*=*p ^{a}qr* and

*n*=

*pqrs*for distinct primes

*p*,

*q*,

*r*,

*s*and positive integer

*a*. In proving our results, we make heavy use of the Strong Perfect Graph Theorem.

#### Sponsor

Mary Marshall, Professor of Mathematics, Illinois College

#### Recommended Citation

Smith, Bennett
(2016)
"Perfect Zero-Divisor Graphs of Z_n,"
*Rose-Hulman Undergraduate Mathematics Journal*: Vol. 17
:
Iss.
2
, Article 6.

Available at:
https://scholar.rose-hulman.edu/rhumj/vol17/iss2/6