Timeline for Graphs associated to the mathematical structures
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 17, 2017 at 6:01 | comment | added | Peter Heinig | @MarianoSuárez-Álvarez: And if I try to survey this field myself, I tend to agree with the ring-theorist: while those graphs are fascinating qua graphs, posing deep problems, and, notably, posing wonderful benchmarks for the state of the art of graph-theory, e.g., the technology availabe to recognize perfect graphs, they don't seem to reflect any respected, traditional, property of commutative rings. | |
| Sep 17, 2017 at 6:00 | comment | added | Peter Heinig | @MarianoSuárez-Álvarez [...] so to speak. What one should note in this context: zero-divisor graphs for non-commutative rings have been defined, though generically they are directed graphs. (Incidentally, ironically, zero-divisor graphs for non-commutative rings seem to have first been defined in a journal having "commutative rings" in its title: [S.P. Redmond, The zero-divisor graph of a non-commutative ring, Internat. J. Commutative Rings 1 (4) (2002) 203–211]. | |
| Sep 17, 2017 at 5:55 | comment | added | Peter Heinig | @MarianoSuárez-Álvarez: yes, the restriction to commutative rings was on purpose. The purpose, though, was rather superficial: to make the comment accurately reflect the conversation I overheard. The ring-theorist was expressly speaking of zero-divisor graphs in commutative rings only. The purpose was not to avoid a natural correspondence (property of a non-commutative ring)$\xrightarrow[]{F}$(property of graphs) under a computable functor $F$ that I knew of and which would contradict my 'claim': I don't know of such a correspondence either. The purpose was empirical, not mathematical, [...] | |
| Sep 16, 2017 at 19:52 | comment | added | Mariano Suárez-Álvarez | @PeterHeinig, is the restriction to commutative rings on purpose? The Gabriel quiver of finite dimensional algebras is quite useful, for example! The Auslandr-Reiten quiver also comes to mind. | |
| Sep 16, 2017 at 9:57 | comment | added | Peter Heinig | @GA316: allow me to add a negative example (or, if you will, a challenge): I recently heard an experienced ring-theorist (which I won't name) say, in the context of discussing notions like zero-divisor-graphs, that he thinks he does not know a single notable example of a graph-theoretic property precisely corresponding to a notable ring-theoretic property, w.r.t. some computable functor $\mathsf{CommutativeRings}\to\mathsf{SomeReasonableCategoryOfGraphs}$. The 'computable' here is important, otherwise one can probably prove something. So there seems work and discoveries left for you... | |
| Sep 15, 2017 at 10:04 | comment | added | GA316 | nice answer. I don't know this. I would like to know more like these kind of theories. If you know any thing more kindly add it to the answer. Thanks. | |
| Sep 15, 2017 at 9:38 | history | answered | coudy | CC BY-SA 3.0 |