Graphs derived from multirings. (English) Zbl 1498.05126
Summary: The purpose of this paper was to introduce the concepts of very thin multigroup, nondistributive (very thin) multirings, zero-divisor elements of multirings and zero-divisor graphs based on zero-divisor elements of multirings. In order to realize the article’s goals, we consider the relationship between finite nondistributive (very thin) multirings and multirings and construct nondistributive (very thin) multirings based on given ring. By some conditions on prime numbers, finite (nondistributive) very thin multirings are constructed. Zero-divisor graph based on zero-divisor set of (nondistributive) (very thin) multirings are introduced, so we investigate of some necessity and sufficiency conditions such that compute of order and size of these zero-divisor graphs. Also, the notations of derivable zero-divisor graphs and derivable zero-divisor subgraphs are introduced and is showed that some multipartite graphs are derivable zero-divisor graphs, all complete graphs, and cyclic graphs are derivable zero-divisor subgraphs.
MSC:
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20N25 | Fuzzy groups |
| 16Y80 | \(\Gamma\) and fuzzy structures |
Keywords:
nondistributive multiring; zero-divisor (sub)graph; very thin multigroup; very thin multiringReferences:
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