#### Document Type

Article

#### Publication Date

3-23-2021

#### Abstract

The zero-divisor graph of a commutative ring R with unity is the graph Γ(R) whose vertex set is the set of nonzero zero divisors of R; where two vertices are adjacent if and only if their product in R is zero. A vertex coloring c : V (G) → Bbb N of a non-trivial connected graph G is called a sigma coloring if σ(u) = σ(ν) for any pair of adjacent vertices u and v. Here; σ(χ) denotes the sum of the colors assigned to vertices adjacent to x. The sigma chromatic number of G; denoted by σ(G); is defined as the least number of colors needed to construct a sigma coloring of G. In this paper; we analyze the structure of the zero-divisor graph of rings Bbb Zn; where n = pn11 P2n2 ...Pmnm; where m,ni,n2; ...,nm are positive integers and p1,p2; ...,pm are distinct primes. The analysis is carried out by partitioning the vertex set of such zero-divisor graphs and analyzing the adjacencies; cardinality; and the degree of the vertices in each set of the partition. Using these properties; we determine the sigma chromatic number of these zero-divisor graphs.

#### Recommended Citation

Garciano, A. D., Marcelo, R. M., Ruiz, M. J. P., & Tolentino, M. A. C. (2021). On the sigma chromatic number of the zero-divisor graphs of the ring of integers modulo n. Journal of Physics: Conference Series, 1836(1), 012013. https://doi.org/10.1088/1742-6596/1836/1/012013