Given a commutative ring R with identity 1?0, let the set Z(R) denote of zero-divisors and Z*(R)=Z(R)?{0} be non-zero R. The zero-divisor graph R, denoted by ?(R), is simple whose vertex Z*(R) each pair vertices in are adjacent when their product 0. In this article, we find structure Laplacian spectrum graphs ?(Zn) for n=pN1qN2, where p<q primes N1,N2 positive integers.

In this article, we determine up to isomorphism of rings, rings R such that R has the following properties: (i) R is a commutative ring with identity which admits at least two nonzero zero-divisors, (ii) the complement of the zero-divisor graph of R is connected and it admits a cut vertex. Indeed, it is proved that there are exactly two such rings up to isomorphism of rings.

In an algebra class, one uses the zero-factor property to solve polynomial equations. For example, consider the equation x 2 = x. Rewriting it as x (x − 1) = 0, we conclude that the solutions are x = 0, 1. However, the same equation in a different number system need not yield the same solutions. For example, in Z 6 (the integers modulo 6), not only 0 and 1, but also 3 and 4 are solutions. (Chec...

In this paper, we introduce a family of graphs which is a generalization of zero-divisor graphs and compute an upper-bound for the diameter of such graphs. We also investigate their cycles and cores

A divisor graph G is an ordered pair (V, E) where V ⊂ and for all u = v ∈ V , uv ∈ E if and only if u | v or v | u. A graph which is isomorphic to a divisor graph is also called a divisor graph. In this note, we will prove that for any n 1 and 0 m n 2 then there exists a divisor graph of order n and size m. We also present a simple proof of the characterization of divisor graphs which is due to...

Let R be a commutative ring and Z(R) the set of all zero divisors R. Γ(R) is said to divisor graph if x,y ∈ V (Γ(R)) = (x,y) E(Γ(R)) only x.y 0. In this paper, we determine total vertex irregularity strength graphs associated with rings ℤp2 × Zq for p,q prime numbers.

Let Γ(Zm) be the zero divisor graph of the ring Zm. In this paper we explore the p-partite structures of Γ(Zm), as well as determine a complete classification of the chromatic number of Γ(Zm). In particular, we explore how these concepts are related to the prime factorization of m.