The total graph and regular graph of a commutative ring

Abstract

Communicated by J. Walker MSC: 05C40 05C45 16P10 16P40a b s t r a c t Let R be a commutative ring. The total graph of R, denoted by T (Γ (R)) is a graph with all elements of R as vertices, and two distinct vertices x, y ∈ R, are adjacent if and only if x + y ∈ Z (R), where Z (R) denotes the set of zero-divisors of R. Let regular graph of R, Reg(Γ (R)), be the induced subgraph of T (Γ (R)) on the regular elements of R. Let R be a commutative Noetherian ring and Z (R) is not an ideal. In this paper we show that if T (Γ (R)) is a connected graph, then diam(Reg(Γ (R))) diam(T (Γ (R))). Also, we prove that if R is a finite ring, then T (Γ (R)) is a Hamiltonian graph. Finally, we show that if S is a commutative Noetherian ring and Reg(S) is finite, then S is finite.

The total zero-divisor graph of a commutative ring

“…Recall that if R is Noetherian ring and M a finitely generated non-zero R-module, then the set of zero-divisors for M is the union of all the associated primes of M [9, Proposition (7.B)]. In the special case where we treat R as an R-module, we have (1) Z(R) = P ∈Ass(R) P.…”

“…To the best of our knowledge, there is no full description of rings with Hamiltonian zero-divisor graphs. In [1] the authors proved that if the total graph of a finite commutative ring is connected then it is also a Hamiltonian graph. In the next proposition we characterize Hamiltonian total zero-divisor graphs and prove that the total zero-divisor graph is Hamiltonian if and only if it is complete with at least 4 vertices.…”

Reducing Redundancy of Codons through Total Graph

“…Anderson and Livingston (1999) associate a graph, Γ(R), to R with vertices Z (R)\{0}, where Z (R) is the set of zerodivisors of R and for distinct x, y ∈ Z (R)\{0}, the vertices x and y are adjacent if and only if xy = 0. Akbari et al (2009) proved that the total graph is a Hamiltonian graph if it is connected.…”

The Total Graphs of Finite Rings

“…They also proved that the total graph of a commutative ring is connected if and only if the set of zero-divisors does not form an ideal. In [1] Akbari et al proved that if the total graph of a finite commutative ring is connected then it is also a Hamiltonian graph. In [6], Maimani et al gave the necessary and sufficient conditions for the total graphs of finite commutative rings to be planar or toroidal and in [8] Tamizh Chelvam and Asir characterized all commutative rings such that their total graphs have genus 2.…”