On the dot product graph of a commutative ring II
In 2015, the second-named author introduced the dot product graph associated to a commutative ring A. Let A be a commutative ring with nonzero identity, 1 ≤ n < ∞ be an integer, and R = A × A × · · · × A (n times). We recall that the total dot product graph of R is the (undirected) graph T D(R) w...
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2020
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| Online Access: | http://hdl.handle.net/11073/18297 |
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| _version_ | 1864513444629184512 |
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| author | Abdulla, Mohammad Ahmad |
| author2 | Badawi, Ayman |
| author2_role | author |
| author_facet | Abdulla, Mohammad Ahmad Badawi, Ayman |
| author_role | author |
| dc.creator.none.fl_str_mv | Abdulla, Mohammad Ahmad Badawi, Ayman |
| dc.date.none.fl_str_mv | 2020-07-15T08:33:29Z 2020-07-15T08:33:29Z 2020 |
| dc.format.none.fl_str_mv | application/pdf |
| dc.identifier.none.fl_str_mv | Abdulla, M. & Badawi, A. (2020). On the dot product graph of a commutative ring II. International Electronic Journal of Algebra, 28, 61-74. doi: 10.24330/ieja.768135 1306-6048 http://hdl.handle.net/11073/18297 10.24330/ieja.7681351306-6048 |
| dc.language.none.fl_str_mv | en_US |
| dc.publisher.none.fl_str_mv | International Electronic Journal of Algebra |
| dc.relation.none.fl_str_mv | https://doi.org/10.24330/ieja.7681351306-6048 |
| dc.subject.none.fl_str_mv | Dot product graph Annihilator graph Total graph Zero-divisor graph |
| dc.title.none.fl_str_mv | On the dot product graph of a commutative ring II |
| dc.type.none.fl_str_mv | Peer-Reviewed Published version info:eu-repo/semantics/publishedVersion info:eu-repo/semantics/article |
| description | In 2015, the second-named author introduced the dot product graph associated to a commutative ring A. Let A be a commutative ring with nonzero identity, 1 ≤ n < ∞ be an integer, and R = A × A × · · · × A (n times). We recall that the total dot product graph of R is the (undirected) graph T D(R) with vertices R∗ = R \ {(0, 0, . . . , 0)}, and two distinct vertices x and y are adjacent if and only if x·y = 0 ∈ A (where x·y denotes the normal dot product of x and y). Let Z(R) denote the set of all zero-divisors of R. Then the zero-divisor dot product graph of R is the induced subgraph ZD(R) of T D(R) with vertices Z(R)∗ = Z(R) \ {(0, 0, . . . , 0)}. Let U(R) denote the set of all units of R. Then the unit dot product graph of R is the induced subgraph UD(R) of T D(R) with vertices U(R). In this paper, we study the structure of T D(R), UD(R), and ZD(R) when A = Zn or A = GF(pn), the finite field with pn elements, where n ≥ 2 and p is a prime positive integer. |
| format | article |
| id | aus_149d8dd15fc7ed05142d7262398d3a83 |
| identifier_str_mv | Abdulla, M. & Badawi, A. (2020). On the dot product graph of a commutative ring II. International Electronic Journal of Algebra, 28, 61-74. doi: 10.24330/ieja.768135 1306-6048 10.24330/ieja.7681351306-6048 |
| language_invalid_str_mv | en_US |
| network_acronym_str | aus |
| network_name_str | aus |
| oai_identifier_str | oai:repository.aus.edu:11073/18297 |
| publishDate | 2020 |
| publisher.none.fl_str_mv | International Electronic Journal of Algebra |
| repository.mail.fl_str_mv | |
| repository.name.fl_str_mv | |
| repository_id_str | |
| spelling | On the dot product graph of a commutative ring IIAbdulla, Mohammad AhmadBadawi, AymanDot product graphAnnihilator graphTotal graphZero-divisor graphIn 2015, the second-named author introduced the dot product graph associated to a commutative ring A. Let A be a commutative ring with nonzero identity, 1 ≤ n < ∞ be an integer, and R = A × A × · · · × A (n times). We recall that the total dot product graph of R is the (undirected) graph T D(R) with vertices R∗ = R \ {(0, 0, . . . , 0)}, and two distinct vertices x and y are adjacent if and only if x·y = 0 ∈ A (where x·y denotes the normal dot product of x and y). Let Z(R) denote the set of all zero-divisors of R. Then the zero-divisor dot product graph of R is the induced subgraph ZD(R) of T D(R) with vertices Z(R)∗ = Z(R) \ {(0, 0, . . . , 0)}. Let U(R) denote the set of all units of R. Then the unit dot product graph of R is the induced subgraph UD(R) of T D(R) with vertices U(R). In this paper, we study the structure of T D(R), UD(R), and ZD(R) when A = Zn or A = GF(pn), the finite field with pn elements, where n ≥ 2 and p is a prime positive integer.International Electronic Journal of Algebra2020-07-15T08:33:29Z2020-07-15T08:33:29Z2020Peer-ReviewedPublished versioninfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfAbdulla, M. & Badawi, A. (2020). On the dot product graph of a commutative ring II. International Electronic Journal of Algebra, 28, 61-74. doi: 10.24330/ieja.7681351306-6048http://hdl.handle.net/11073/1829710.24330/ieja.7681351306-6048en_UShttps://doi.org/10.24330/ieja.7681351306-6048oai:repository.aus.edu:11073/182972024-08-22T12:02:17Z |
| spellingShingle | On the dot product graph of a commutative ring II Abdulla, Mohammad Ahmad Dot product graph Annihilator graph Total graph Zero-divisor graph |
| status_str | publishedVersion |
| title | On the dot product graph of a commutative ring II |
| title_full | On the dot product graph of a commutative ring II |
| title_fullStr | On the dot product graph of a commutative ring II |
| title_full_unstemmed | On the dot product graph of a commutative ring II |
| title_short | On the dot product graph of a commutative ring II |
| title_sort | On the dot product graph of a commutative ring II |
| topic | Dot product graph Annihilator graph Total graph Zero-divisor graph |
| url | http://hdl.handle.net/11073/18297 |