On the Sigma Chromatic Number of the Zero-Divisor Graphs of the Ring of Integers Modulo n

• Main Authors: Garciano, Agnes, Marcelo, Reginaldo M, Ruiz, Mari-Jo P, Tolentino, Mark Anthony C
• Format: text
• Published: Archīum Ateneo 2021
• Subjects: zero-divisor graphs; sigma coloring; Mathematics
• Online Access: https://archium.ateneo.edu/mathematics-faculty-pubs/161; https://archium.ateneo.edu/cgi/viewcontent.cgi?article=1164&context=mathematics-faculty-pubs
• Institution: Ateneo De Manila University

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.