CoCalc Public FilesStable graphs.ipynbOpen in with one click!
Author: Johannes Schmitt
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Description: An example file for computations with stable graphs in the package admcycles.

Using admcycles it is possible to compute lists of stable graphs using the function list_strata(g,n,e). It returns a list of isomorphism classes of stable graphs in genus g with n legs and e edges.

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from admcycles import * # loads the package admcycles L = list_strata(1,2,1) # computes list of stable graphs with g=1, n=2 and e=1 edge for Gamma in L: print(Gamma) # prints all graphs

A stable graph is represented by three lists:

  • a list of the genera of the vertices
  • for each vertex the list of half-edges incident to the vertex
  • a list of pairs of half-edges which are connected by an edge; half-edges named 1,...,n are the legs of the graph

So the stable graph [0, 1] [[1, 2, 3], [4]] [(3, 4)] has two vertices (of genus 0 and 1, respectively). The genus 0 vertex has half-edges 1,2,3 and the genus 1 vertex has a half-edge 4. The half-edges 3,4 form an edge.

In the standard language of stable graphs you would say that the graph above is given by

  • V = {v_1, v_2}, g(v_1) = 0, g(v_2) = 1
  • H = {1,2,3,4}, v(1) = v_1, v(2) = v_1, v(3) = v_1, v(4) = v_2
  • L = {1,2}, \ell : L \to {1,2}, \ell(1) = 1, \ell(2) = 2
  • ι\iota : H \to H, ι\iota(1)=1, ι\iota(2)=2, ι\iota(3)=4, ι\iota(4)=3

We can also input a stable graph by giving three lists as above, for instance to compute the number of its automorphisms.

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Gamma = StableGraph([1,1,1],[[1,6],[2,3],[4,5]],[(1,2),(3,4),(5,6)]) # Gamma = triangle of vertices of genus 1 Gamma.automorphism_number() # Aut(Gamma) = ?

If we want to list all strata of a given genus and number of markings, we can use the following function.

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def all_strata(g,n): return sum((list_strata(g,n,e) for e in range(0,3*g-3+n+1)),[]) # Exercise (if you know Python): Understand why the above function does what we want. L = all_strata(1,2) for Gamma in L: print(Gamma) # prints all graphs

One thing we can check is that the number of stable graphs grows extremely fast as g,n increase.

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for n in range(1,6): print('The number of stable graphs with g=1, n='+repr(n)+' is equal to '+repr(len(all_strata(1,n))))
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