In this paper we consider, for a finite commutative ring R, the wellstudied zero-divisor graph Γ(R) and the compressed zero-divisor graph Γc(R) of R and a newly-defined graphical structure — the zero-divisor lattice Λ(R) of R. We give results which provide information when Γ(R) ∼= Γ(S), Γc(R) ∼= Γc(S), and Λ(R) ∼= Λ(S) for two finite commutative rings R and S. We also provide a theorem which sa...

Inspired by a very recent work of A. Đurić, S. Jevđenić and N. Stopar, we introduce new definition zero-divisor graphs attached to rings that includes all the classical definitions already known in literature. We provide an interpretation such means functor call which is associated with family special equivalence relations fixed beforehand. thus recover generalize many results for framework mig...

The concept of a zero-divisor graph of a commutative ring was first introduced in Beck (1988), and later redefined in Anderson and Livingston (1999). Redmond (2002) further extended this concept to the noncommutative case, introducing several definitions of a zero-divisor graph of a noncommutative ring. Recently, the diameter and girth of polynomial and power series rings over a commutative rin...

‎the concept of the bipartite divisor graph for integer subsets has been considered in [‎m‎. ‎a‎. ‎iranmanesh and c‎. ‎e‎. ‎praeger‎, ‎bipartite divisor graphs for integer subsets‎, ‎{em graphs combin.}‎, ‎{bf 26} (2010) 95--105‎.]‎. ‎in this paper‎, ‎we will consider this graph for the set of character degrees of a finite group $g$ and obtain some properties of this graph‎. ‎we show that if $g...

Let $R$ be a ring with unity. The undirected nilpotent graph of $R$, denoted by $Gamma_N(R)$, is a graph with vertex set ~$Z_N(R)^* = {0neq x in R | xy in N(R) for some y in R^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in N(R)$, or equivalently, $yx in N(R)$, where $N(R)$ denoted the nilpotent elements of $R$. Recently, it has been proved that if $R$ is a left A...

Let $A$ be a commutative ring with nonzero identity, and $1leq n

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...