implement the magnitude function of a graph

[chapoton]

the magnitude function is some kind of generalized Euler characteristic defined for metric spaces

in particular, graphs can be seen as metric spaces with edges of length one.

Let us implement the magnitude function for graphs.

[chapoton]

branch not yet ready for review

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New commits:

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implement the magnitude function of a graph

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

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trac 21308 better doc

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

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trac 21308 add one reference + more doc

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​87f18ad

trac 21308 better doc

[jmantysalo]

Never heard about this before... Anyways, at ​https://www.math.uni-bielefeld.de/documenta/vol-18/27.pdf p. 875 there is an example of magnitude function for 2+3 complete bipartite graph. It seems to go to value 5 when t grows. However

sage: g = graphs.CompleteBipartiteGraph(3, 2)
sage: m = g.magnitude_function()
sage: m(4).n(20)
0.14839
sage: m(10).n(20)
0.029694

I suppose that I understood something wrong.

[chapoton]

What is pictured page 875 is the function "magnitude in the variable t" defined by replacing q by exp(-t)

Saying that the limit of that when t tends to +infinity is the cardinality, is the same as saying that the value of the fraction at q=0 is the cardinality.

[jmantysalo]

OK then. I would write "substitution q = e^{-t} to obtain" instead of "substitution q = exp(-t) to obtain", but not that big thing.

If I understand this correctly, it is much faster to compute by first substituting t and then inverting the resulting numerical matrix --- assuming the user wants only few values. However, this implementation is propably what the user wants.

Tests passed, documentation builds etc, so I mark this as positive review.