For an arbitrary ring $R$, the zero-divisor graph of $R$, denoted by $Gamma (R)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $R$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. It is well-known that for any commutative ring $R$, $Gamma (R) cong Gamma (T(R))$ where $T(R)$ is the (total) quotient ring of $R$. In this...

‎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...

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

‎In this paper we give a characterization for all commutative‎ ‎rings with $1$ whose zero-divisor graphs are $C_4$-free.‎

for an arbitrary ring $r$, the zero-divisor graph of $r$, denoted by $gamma (r)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $r$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. it is well-known that for any commutative ring $r$, $gamma (r) cong gamma (t(r))$ where $t(r)$ is the (total) quotient ring of $r$. in this...

Number theoretic graphs are one of the emerging fields in Graph theory. This article is a study on existing research results graphs, especially Divisor Graphs (DG)s, Function (DFGs) and Cayley (DCGs). We have provided brief survey benchmark findings regarding above-mentioned graphs.

‎in this paper we give a characterization for all commutative‎ ‎rings with $1$ whose zero-divisor graphs are $c_4$-free.‎

In this paper we study zero-divisor graphs of semirings. We show that all zero-divisor graphs of (possibly noncommutative) semirings are connected and have diameter less than or equal to 3. We characterize all acyclic zero-divisor graphs of semirings and prove that in the case zero-divisor graphs are cyclic, their girths are less than or equal to 4. We also give a description of the zero-diviso...

In this paper we study sub-semigroups of a zero-divisor semigroup S determined by properties of the zero-divisor graph Γ(S). We use these sub-semigroups to study the correspondence between zero-divisor semigroups and zero-divisor graphs. We study properties of sub-semigroups of Boolean semigroups via the zero-divisor graph. As an application, we provide a characterization of the graphs which ar...