Distances in zero-divisor and total graphs from commutative rings–A survey
There are so many ways to construct graphs from algebraic structures. Most popular constructions are Cayley graphs, commuting graphs and non-commuting graphs from finite groups and zero-divisor graphs and total graphs from commutative rings. For a commutative ring R with non-zero identity, we denote...
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| Format: | Article |
| Language: | English |
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Taylor & Francis Group
2016-12-01
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| Series: | AKCE International Journal of Graphs and Combinatorics |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S0972860016301918 |
| _version_ | 1818275366419038208 |
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| author | T. Tamizh Chelvam T. Asir |
| author_facet | T. Tamizh Chelvam T. Asir |
| author_sort | T. Tamizh Chelvam |
| collection | DOAJ |
| description | There are so many ways to construct graphs from algebraic structures. Most popular constructions are Cayley graphs, commuting graphs and non-commuting graphs from finite groups and zero-divisor graphs and total graphs from commutative rings. For a commutative ring R with non-zero identity, we denote the set of zero-divisors and unit elements of R by Z(R) and U(R), respectively. One of the associated graphs to a ring R is the zero-divisor graph; it is a simple graph with vertex set Z(R)∖{0}, and two vertices x and y are adjacent if and only if xy=0. This graph was first introduced by Beck, where all the elements of R are considered as the vertices. Anderson and Badawi, introduced the total graph of R, as the simple graph with all elements of R as vertices, and two distinct vertices x and y are adjacent if and only if x+y∈Z(R). For a given graph G, the concept of connectedness, diameter and girth are always of great interest. Several authors extensively studied about the zero-divisor and total graphs from commutative rings. In this paper, we present a survey of results obtained with regard to distances in zero-divisor and total graphs. |
| first_indexed | 2024-12-12T22:28:37Z |
| format | Article |
| id | doaj.art-5fd1d776e2f8469b8022289b60cfd8d1 |
| institution | Directory Open Access Journal |
| issn | 0972-8600 |
| language | English |
| last_indexed | 2024-12-12T22:28:37Z |
| publishDate | 2016-12-01 |
| publisher | Taylor & Francis Group |
| record_format | Article |
| series | AKCE International Journal of Graphs and Combinatorics |
| spelling | doaj.art-5fd1d776e2f8469b8022289b60cfd8d12022-12-22T00:09:41ZengTaylor & Francis GroupAKCE International Journal of Graphs and Combinatorics0972-86002016-12-0113329029810.1016/j.akcej.2016.11.009Distances in zero-divisor and total graphs from commutative rings–A surveyT. Tamizh Chelvam0T. Asir1Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli 627012, Tamil Nadu, IndiaDepartment of Mathematics(DDE), Madurai Kamaraj University, Madurai 625 021, Tamil Nadu, IndiaThere are so many ways to construct graphs from algebraic structures. Most popular constructions are Cayley graphs, commuting graphs and non-commuting graphs from finite groups and zero-divisor graphs and total graphs from commutative rings. For a commutative ring R with non-zero identity, we denote the set of zero-divisors and unit elements of R by Z(R) and U(R), respectively. One of the associated graphs to a ring R is the zero-divisor graph; it is a simple graph with vertex set Z(R)∖{0}, and two vertices x and y are adjacent if and only if xy=0. This graph was first introduced by Beck, where all the elements of R are considered as the vertices. Anderson and Badawi, introduced the total graph of R, as the simple graph with all elements of R as vertices, and two distinct vertices x and y are adjacent if and only if x+y∈Z(R). For a given graph G, the concept of connectedness, diameter and girth are always of great interest. Several authors extensively studied about the zero-divisor and total graphs from commutative rings. In this paper, we present a survey of results obtained with regard to distances in zero-divisor and total graphs.http://www.sciencedirect.com/science/article/pii/S0972860016301918Commutative ringZero-divisor graphTotal graphDomination |
| spellingShingle | T. Tamizh Chelvam T. Asir Distances in zero-divisor and total graphs from commutative rings–A survey AKCE International Journal of Graphs and Combinatorics Commutative ring Zero-divisor graph Total graph Domination |
| title | Distances in zero-divisor and total graphs from commutative rings–A survey |
| title_full | Distances in zero-divisor and total graphs from commutative rings–A survey |
| title_fullStr | Distances in zero-divisor and total graphs from commutative rings–A survey |
| title_full_unstemmed | Distances in zero-divisor and total graphs from commutative rings–A survey |
| title_short | Distances in zero-divisor and total graphs from commutative rings–A survey |
| title_sort | distances in zero divisor and total graphs from commutative rings a survey |
| topic | Commutative ring Zero-divisor graph Total graph Domination |
| url | http://www.sciencedirect.com/science/article/pii/S0972860016301918 |
| work_keys_str_mv | AT ttamizhchelvam distancesinzerodivisorandtotalgraphsfromcommutativeringsasurvey AT tasir distancesinzerodivisorandtotalgraphsfromcommutativeringsasurvey |