Perfect Zero-Divisor Graphs of 𝕫ₙ

Authors

Bennett Smith, Illinois College

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^aqr 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.

Author Bio

Bennett Smith was a mathematics and physics double major who graduated from Illinois College in May of 2016. He began his undergraduate research after taking an abstract algebra course. From the course, he knew that he wanted to learn more about the field and began looking for potential research topics. One thing he enjoys about both mathematics and physics is that many problems can be solved via computer programming, a hobby that he actively engages in and employed to aid in his research. Bennett is now planning to work in industry for a few years before starting graduate studies.

Faculty Sponsor

Mary Marshall

Recommended Citation

Smith, Bennett (2016) "Perfect Zero-Divisor Graphs of 𝕫ₙ," Rose-Hulman Undergraduate Mathematics Journal: Vol. 17: Iss. 2, Article 6.
Available at: https://scholar.rose-hulman.edu/rhumj/vol17/iss2/6