The total graph and regular graph of a commutative ring
By Saieed Akbari, Dariush Kiani, F. Mohammadi, Somayeh Moradi. 2009
138 citations recorded by the source corpus.
Abstract 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 , R e g ( 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 ( R e g ( 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 R e g ( S ) is finite, then S is finite.
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/192cdb8f187967ea8ececef5b221922d1464856e
How researchers see this paper
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In Algebra and Number Theory
- Ayman Badawi
- David F. Anderson
- Thangaraj Asir
- T. Tamizh Chelvam
- M. Behboodi
- Saieed Akbari
- F. Soheilnia
- Ahmad Yousefian Darani
In Geometry and Topology
- Ayman Badawi
- Thangaraj Asir
- T. Tamizh Chelvam
- David F. Anderson
- M. Behboodi
- Saieed Akbari
- Mohammad Javad Nikmehr
Citations in the corpus
- The Classification of the Annihilating-Ideal Graphs of Commutative Rings (2014, background): “There are many papers on assigning a graph to a ring (for instance, see [1, 2, 4, 12, 13, 16]).”
- On the Total Graph and Its Complement of a Commutative Ring (2013, methodology): “From this and by Lemma 1 [1], we get that R 4 2 x / x2 3 × r2 4 × r2 2 x / x2 × r2 where r is a natural number (not necessarily finite).”
- ON THE TOTAL GRAPH OF A COMMUTATIVE RING WITHOUT THE ZERO ELEMENT (2012, background): “The total graph has also been investigated in [1, 2, 14].”
- On the Genus of the Total Graph of a Commutative Ring (2013, background): “After various research articles have been published on the total graph of a commutative ring [1, 13, 17, 18].”