Zero Divisor Graph of Quotient Ring

Abstract

Recently, a lot of research has been carried out regarding graphs built from algebraic structures, including ring structures. One important example of a graph constructed from a ring is the zero divisor graph. For a commutative ring R, the zero divisor graph Γ(R) is defined as a simple graph with vertices that are non-zero zero divisors of R, and two distinct vertices are adjacent if and only if the product of the vertices is equal to zero. In this paper, we investigate the zero divisor graph of the quotient ring ℤp[x]/⟨x5⟩ with prime p. More precisely, we characterize some graph properties, including the order, size, adjacency matrix, degree, distance, diameter, girth, clique number, and chromatic number of Γ(ℤp[x]/⟨x5⟩).

Keywords

• commutative ring
• quotient ring
• graphs properties
• zero divisor graph.