This paper deals with some results concerning the notion of extended ideal based zero divisor graph $overline Gamma_I(R)$ for an ideal $I$ of a commutative ring $R$ and characterize its bipartite graph. Also, we study the properties of an annihilator of $overline Gamma_I(R)$.

in this article, we give several generalizations of the concept of annihilating ideal graph over a commutative ring with identity to modules. weobserve that over a commutative ring $r$, $bbb{ag}_*(_rm)$ isconnected and diam$bbb{ag}_*(_rm)leq 3$. moreover, if $bbb{ag}_*(_rm)$ contains a cycle, then $mbox{gr}bbb{ag}_*(_rm)leq 4$. also for an $r$-module $m$ with$bbb{a}_*(m)neq s(m)setminus {0}$, $...

In this paper, we show that Q n is a divisor graph, for n = 2, 3. For n ≥ 4, we show that Q n is a divisor graph iff k ≥ n− 1. For folded-hypercube, we get FQn is a divisor graph when n is odd. But, if n ≥ 4 is even integer, then FQn is not a divisor graph. For n ≥ 5, we show that (FQn) k is not a divisor graph, where 2 ≤ k ≤ ⌈ 2 ⌉ − 1.

let $m$ be an $r$-module and $0 neq fin m^*={rm hom}(m,r)$. we associate an undirected graph $gf$ to $m$ in which non-zero elements $x$ and $y$ of $m$ are adjacent provided that $xf(y)=0$ or $yf(x)=0$. weobserve that over a commutative ring $r$, $gf$ is connected anddiam$(gf)leq 3$. moreover, if $gamma (m)$ contains a cycle,then $mbox{gr}(gf)leq 4$. furthermore if $|gf|geq 1$, then$gf$ is finit...

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 $i$ be a proper ideal of a commutative semiring $r$ and let $p(i)$ be the set of all elements of $r$ that are not prime to $i$. in this paper, we investigate the total graph of $r$ with respect to $i$, denoted by $t(gamma_{i} (r))$. it is the (undirected) graph with elements of $r$ as vertices, and for distinct $x, y in r$, the vertices $x$ and $y$ are adjacent if and only if $x + y in p(i)...

In this paper, we prove that for any positive integers k, n with k ≥ 2, the graph P k n is a divisor graph if and only if n ≤ 2k + 2, where P k n is the k power of the path Pn. For powers of cycles we show that C n is a divisor graph when n ≤ 2k + 2, but is not a divisor graph when n ≥ 2k + bk2 c+ 3, where C k n is the k th power of the cycle Cn. Moreover, for odd n with 2k + 2 < n < 2k + bk2 c...

A divisor cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2,... | |} V such that an edge uv is assigned the label 1 if either ( ) | ( ) f u f v or ( ) | ( ) f v f u and the label 0 if ( ) ( ) f u f v , then number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph with a divisor cordial labeling is called a divisor cordial ...

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...

The zero-divisor graph of a commutative semigroup with zero is the graph whose vertices are the nonzero zero-divisors of the semigroup, with two distinct vertices adjacent if the product of the corresponding elements is zero. New criteria to identify zerodivisor graphs are derived using both graph-theoretic and algebraic methods. We find the lowest bound on the number of edges necessary to guar...