The total graph and regular graph of a commutative ring |
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Authors: | S Akbari D Kiani F Mohammadi |
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Institution: | a Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran b Department of Pure Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Iran c School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran |
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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, 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 . 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. |
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Keywords: | 05C40 05C45 16P10 16P40 |
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