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Is the inequality in an ultrametric space always an equality when ? GeoffreyT2000 ( talk) 02:44, 9 September 2016 (UTC)
- Yes. According to the article (and it's straightforward to prove), for any three points x, y, z, at least one of d(x,y)=d(x,z), d(x,y)=d(y,z), d(x,z)=d(y,z) is true. Equivalently, if d(x,y)≠d(y,z) then d(x,z) must be either d(x,y) or d(y,z). Suppose the first case, d(x,z)=d(x,y). Then d(y,z)≤max(d(x,z), d(x,y)) = d(x,y), max(d(x,y), d(y,z))=d(x,y)=d(x,z) as required. The other case is similar. -- RDBury ( talk) 08:10, 9 September 2016 (UTC)
In graph theory we have the following inequality: connectivity <= edge connectivity <= minimum vertex degree of the graph. You can't improve on this inequality: you can construct a graph with any set of three integers that satisfy it. How would you describe this type of inequality that can't be improved on? 24.255.17.182 ( talk) 22:50, 9 September 2016 (UTC)
- Is that the answer? The property being discussed, described in
Connectivity (graph theory)#Bounds on connectivity, is that κ(G) ≤ λ(G) ≤ δ(G) for any graph G. The fact that there exists some graph G with κ(G) = λ(G) = δ(G) is sufficient for the inequality to be sharp. But isn't our questioner going beyond this and stating that for any positive integers k, l, d where k ≤ l ≤ d, a graph G can be constructed such that κ(G) = k, λ(G) = l, δ(G) = d, and asking for a name for such an inequality? --
ToE
12:51, 10 September 2016 (UTC)
- Sharp inequality sounds good to me. Dmcq ( talk) 22:45, 10 September 2016 (UTC)
- Is that the answer? The property being discussed, described in
Connectivity (graph theory)#Bounds on connectivity, is that κ(G) ≤ λ(G) ≤ δ(G) for any graph G. The fact that there exists some graph G with κ(G) = λ(G) = δ(G) is sufficient for the inequality to be sharp. But isn't our questioner going beyond this and stating that for any positive integers k, l, d where k ≤ l ≤ d, a graph G can be constructed such that κ(G) = k, λ(G) = l, δ(G) = d, and asking for a name for such an inequality? --
ToE
12:51, 10 September 2016 (UTC)
| Mathematics desk | ||
|---|---|---|
| < September 8 | << Aug | September | Oct >> | September 10 > |
| Welcome to the Wikipedia Mathematics Reference Desk Archives |
|---|
| The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Is the inequality in an ultrametric space always an equality when ? GeoffreyT2000 ( talk) 02:44, 9 September 2016 (UTC)
- Yes. According to the article (and it's straightforward to prove), for any three points x, y, z, at least one of d(x,y)=d(x,z), d(x,y)=d(y,z), d(x,z)=d(y,z) is true. Equivalently, if d(x,y)≠d(y,z) then d(x,z) must be either d(x,y) or d(y,z). Suppose the first case, d(x,z)=d(x,y). Then d(y,z)≤max(d(x,z), d(x,y)) = d(x,y), max(d(x,y), d(y,z))=d(x,y)=d(x,z) as required. The other case is similar. -- RDBury ( talk) 08:10, 9 September 2016 (UTC)
In graph theory we have the following inequality: connectivity <= edge connectivity <= minimum vertex degree of the graph. You can't improve on this inequality: you can construct a graph with any set of three integers that satisfy it. How would you describe this type of inequality that can't be improved on? 24.255.17.182 ( talk) 22:50, 9 September 2016 (UTC)
- Is that the answer? The property being discussed, described in
Connectivity (graph theory)#Bounds on connectivity, is that κ(G) ≤ λ(G) ≤ δ(G) for any graph G. The fact that there exists some graph G with κ(G) = λ(G) = δ(G) is sufficient for the inequality to be sharp. But isn't our questioner going beyond this and stating that for any positive integers k, l, d where k ≤ l ≤ d, a graph G can be constructed such that κ(G) = k, λ(G) = l, δ(G) = d, and asking for a name for such an inequality? --
ToE
12:51, 10 September 2016 (UTC)
- Sharp inequality sounds good to me. Dmcq ( talk) 22:45, 10 September 2016 (UTC)
- Is that the answer? The property being discussed, described in
Connectivity (graph theory)#Bounds on connectivity, is that κ(G) ≤ λ(G) ≤ δ(G) for any graph G. The fact that there exists some graph G with κ(G) = λ(G) = δ(G) is sufficient for the inequality to be sharp. But isn't our questioner going beyond this and stating that for any positive integers k, l, d where k ≤ l ≤ d, a graph G can be constructed such that κ(G) = k, λ(G) = l, δ(G) = d, and asking for a name for such an inequality? --
ToE
12:51, 10 September 2016 (UTC)