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...

Let R be a commutative ring with identity and A(R) be the set of ideals with nonzero annihilator. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R) = A(R)r {0} and two distinct vertices I and J are adjacent if and only if IJ = 0. In this paper, we study the domination number of AG(R) and some connections between the domination numbers of annihilating-ideal...

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...

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...

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...

Abstract. A sum divisor cordial labeling of a graph G with vertex set V is a bijection f from V to {1,2,...,| ( ) |} V G such that each edge uv assigned the label 1 if 2 divides ( ) ( ) f u f v + and 0 otherwise. Further, the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph with a sum divisor cordial labeling is called a sum divisor cordial grap...