In this article we discuss the graphs of the sets of zero-divisors of a ring. Now let R be a ring. Let G be a graph with elements of R as vertices such that two non-zero elements a, b ∈ R are adjacent if ab = ba = 0. We examine such a graph and try to find out when such a graph is planar and when is it complete etc. Mathematics Subject Classification: Primary 16-xx, 05-xx; Secondary 05C50

Let N be a near-ring. In this paper, we associate a graph corresponding to the 3-prime radical I of N , denoted by ΓI(N). Further we obtain certain topological properties of Spec(N), the spectrum of 3-prime ideals of N and graph theoretic properties of ΓI(N). Using these properties, we discuss dominating sets and connected dominating sets of ΓI(N).

One of the basic facts of group theory is that each finite group contains a Sylow p-subgroup for each prime p which divides the order of the group. In this note we show that each vertex-transitive selfcomplementary graph has an analogous property. As a consequence of this fact, we obtain that each prime divisor p of the order of a vertex-transitive self-complementary graph satisfies the congrue...

Abstract In this paper, we give a constructive proof of the fact that treewidth graph is at most its divisorial gonality. The gives polynomial time algorithm to construct tree decomposition width k , when an effective divisor degree reaches all vertices given. We also similar result for two related notions: stable gonality and

Consider the (p,q) simple connected graph . The sum absolute values of spectrum quotient matrix a make up graph's energy. objective this study is to examine energy identity graphs and zero-divisor commutative rings using group theory, applications. In study, derived from few classes ring R are examined.

Let Γ(Zm) be the zero divisor graph of the ring Zm. In this paper we explore the p-partite structures of Γ(Zm), as well as determine a complete classification of the chromatic number of Γ(Zm). In particular, we explore how these concepts are related to the prime factorization of m.

Let $R$ be a commutative ring without identity. The zero-divisor graph of $R,$ denoted by $\Gamma(R)$ is with vertex set $Z(R)\setminus \{0\}$ which the all nonzero elements and two distinct vertices $x$ $y$ are adjacent if only $xy=0.$ In this paper, we characterize rings whose graphs outerplanar graphs. Further, establish planar index, index finite

Let R be a commutative ring with identity and let I be an ideal of R. Let R 1 I be the subring of R×R consisting of the elements (r,r + i) for r ∈ R and i ∈ I. We study the diameter and girth of the zero-divisor graph of the ring R 1 I.

Rank of divisor on graph was introduced in 2007 and it quickly attracts many attentions. Recently in 2015, the problem of computing this quantity was proved to be NP-hard. In this paper, we describe a linear time algorithm for this problem limited on cactus graphs.