The zero-divisor graph Π(R) of a commutative ring R is the whose vertices are elements such that u and v adjacent if only uv = 0. If graphs G H have same chromatic polynomial, then we say they chromatically equivalent (or χ−equivalent), written as ∼ H. Suppose uniquely determined by its polynomial. Then it said to be unique χ-unique). In this paper, discuss question: For which numbers n Π(Zn) χ...

AdigraphC is called a zero divisor if there exist non-isomorphic digraphs A and B for which A×C ∼= B ×C , where the operation is the direct product. In other words,C being a zero divisormeans that cancellation property A×C ∼= B×C ⇒ A ∼= B fails. Lovász proved that C is a zero divisor if and only if it admits a homomorphism into a disjoint union of directed cycles of prime lengths.Thus any digra...

This paper investigates properties of the zero-divisor graph of a commutative ring and its genus. In particular, we determine all isomorphism classes of finite commutative rings with identity whose zero-divisor graph has

A (finite or infinite) complete bipartite graph together with some end vertices all adjacent to a common vertex is called a complete bipartite graph with a horn. For any bipartite graph G, we show that G is the graph of a commutative semigroup with 0 if and only if it is one of the following graphs: star graph, two-star graph, complete bipartite graph, complete bipartite graph with a horn. We a...

The study of zero-divisors in group rings had become interesting problem since 1940 with the famous zero-divisor conjecture proposed by G.Higman [2]. Since then several researchers [1, 2, 3] have given partial solutions to this conjecture. Till date the problem remains unsolved. Now we introduce the notions of Smarandache zero divisors (S-zero divisors) and Smarandache week zero divisors (S-wea...

The idea of associating a graph with the zero-divisors of a commutative ring was originated by Beck. The problems concerning zero-divisor graphs have been studied extensively in the past 10 years. DeMeyer and DeMeyer presented some properties for the correspondence between zero-divisor graphs and their semigroups. It is very important to have adequate examples before the complete resolution of ...

Let $\mathcal C_{n}$ be the Catalan monoid on $X_{n}=\{1,\ldots ,n\}$ under its natural order. In this paper, we describe sets of left zero-divisors, right zero-divisors and two sided C_{n}$; their numbers. For $n \geq 4$, define an undirected graph $\Gamma(\mathcal C_{n})$ associated with whose vertices are excluding zero element $\theta$ distinct $\alpha$ $\beta$ joined by edge in case $\alph...