ON THE TOTAL GRAPH OF A COMMUTATIVE RING WITHOUT THE ZERO ELEMENT
2012 ◽
Vol 11
(04)
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pp. 1250074
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Keyword(s):
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)).
Keyword(s):
2019 ◽
Vol 19
(05)
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pp. 2050089
Keyword(s):
2019 ◽
Vol 18
(10)
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pp. 1950190
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Keyword(s):
2015 ◽
Vol 07
(01)
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pp. 1550004
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Keyword(s):
2011 ◽
Vol 03
(04)
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pp. 413-421
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Keyword(s):
2014 ◽
Vol 13
(07)
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pp. 1450047
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Keyword(s):
2015 ◽
Vol 14
(06)
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pp. 1550079
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Keyword(s):