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ON THE TOTAL GRAPH OF A COMMUTATIVE RING WITHOUT THE ZERO ELEMENT

Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1320, USA

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Department of Mathematics and Statistics, American University of Sharjah, P. O. Box 26666, Sharjah, United Arab Emirates

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https://doi.org/10.1142/S0219498812500740Cited by:34 (Source: Crossref)

Abstract

Let R be a commutative ring with nonzero identity, and let Z(R) be its set of zero-divisors. The total graph of R is the (undirected) graph T(Γ(R)) with vertices all elements of R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). In this paper, we study the two (induced) subgraphs Z₀(Γ(R)) and T₀(Γ(R)) of T(Γ(R)), with vertices Z(R)\{0} and R\{0}, respectively. We determine when Z₀(Γ(R)) and T₀(Γ(R)) are connected and compute their diameter and girth. We also investigate zero-divisor paths and regular paths in T₀(Γ(R)).

Keywords:

AMSC: 13A15, 05C99

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