In this paper, we study some operations which produce new divisor graphs from old ones. We prove that the contraction of a divisor graph along a bridge is a divisor graph. For two transmitters (receivers) u and v in some divisor orientation of a divisor graph G, it is shown that the merger G |u,v is also a divisor graph. Two special types of vertex splitting are introduced, one of which produce...

A topological index is a numeric quantity associated with chemical structure that attempts to link the various physicochemical properties, reactivity, or biological activity. Let R be commutative ring identity, and Z*(R) set of all non-zero zero divisors R. Then, Γ(R) said zero-divisor graph if only a·b=0, where a,b∈V(Γ(R))=Z*(R) (a,b)∈E(Γ(R)). We define a∼b a·b=0 a=b. ∼ always reflexive symmet...

Let G = (V, E) be a graph with p vertices and q edges. A has product-set labeling if there exist an injective function f: V(G) → P (N) such that the induced edge f∗: P(N) is defined as f∗ (μν) f (μ) ∗ (ν) ∀μ, ν ∈ E(G).f {ab: (μ), b }. In this paper we investigate how works for zero-divisor line of graphs.

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...

let r be a fnite commutative ring and n(r) be the set of non unit elements of r. the non unit graph of r, denoted by gamma(r), is the graph obtained by setting all the elements of n(r) to be the vertices and defning distinct vertices x and y to be adjacent if and only if x - yin n(r). in this paper, the basic properties of gamma(r) are investigated and some characterization results regarding co...

We define a subgraph of the zero divisor graph of a ring, associated to the ring idempotents. We study its properties and prove that for large classes of rings the connectedness of the graph is equivalent to the indecomposability of the ring and in those cases we also calculate the graph’s diameter. 2000 Mathematics subject classification: 16U99, 05C99.

Critical to the understanding of a graph are its chromatic number and whether or not it is perfect. Here we prove when Γ(Zn), the zero-divisor graph of Zn, is perfect and show an alternative method to [D] for determining the chromatic number in those cases. We go on to determine the chromatic number for Γ(Zp[x]/〈x 〉) where p is prime and show that an isomorphism exists between this graph and Γ(...

In this paper we will investigate the interactions between the zero divisor graph, the annihilator class graph, and the associate class graph of commutative rings. Acknowledgements: We would like to thank the Center for Applied Mathematics at the University of St. Thomas for funding our research. We would also like to thank Dr. Michael Axtell for his help and guidance, as well as Darrin Weber f...