On Mon, May 18, 2015 at 7:51 AM, Vincent Delecroix
<20100.delecr...@gmail.com> wrote:
> Moreover, for non oriented graph the Sage definition does not fit with
> wikipedia... I opened the trac ticket
>
> http://trac.sagemath.org/ticket/18440
>
Thank you.
In case it's of any interest, the code I've been using privately
allows for an arbitrary edge orientation (following Biggs). See below:
def incidence_value(Gamma, v, e, eo):
"""
This computes the incidence value of a vertex and edge of
a graph Gamma with edge-orientation vector eo.
INPUT:
Gamma - graph
v - vertex of Gamma
e - edge of Gamma
eo - a vector of 1's and -1's whose length is the number of
edges in Gamma
EXAMPLES:
sage: Gamma = graphs.PaleyGraph(9)
sage: E = Gamma.edges()
sage: V = Gamma.vertices()
sage: eo = [1]*len(E)
sage: incidence_value(Gamma, V[2], E[3], eo)
0
sage: incidence_value(Gamma, V[8], E[3], eo)
-1
"""
E = Gamma.edges()
if v in e:
if v == e[0]:
k = E.index(e)
return eo[k]
elif v == e[1]:
k = E.index(e)
return -eo[k]
else:
return 0
return 0
def incidence_matrix(Gamma, eo):
"""
This computes the incidence matrix (whose rows are indexed by edges
and whose columns are indexed by vertices) of a graph Gamma with
edge-orientation vector eo. The ordering of the edges and of the
vertices is the same
as Sage's vertices and edges methods.
INPUT:
Gamma - graph
eo - a vector of 1's and -1's whose length is the number of
edges in Gamma
(ie, the size of Gamma, M)
EXAMPLES:
sage: Gamma = graphs.PaleyGraph(9)
sage: E = Gamma.edges()
sage: V = Gamma.vertices()
sage: eo = [1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1,
1, 1, -1, -1]
sage: B = incidence_matrix(Gamma, eo); B
[ 1 -1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[-1 0 0 0 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0]
[ 0 1 0 0 1 0 0 -1 1 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 1 0 1 -1 -1 0 0 0 0 0 0]
[ 0 0 -1 0 0 0 0 0 0 -1 0 0 1 -1 0 0 0 0]
[ 0 0 0 0 0 1 0 0 0 0 1 0 -1 0 1 0 0 0]
[ 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 1 -1 0]
[ 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 -1 0 -1]
[ 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 1 1]
sage: B.transpose()*B == Gamma.laplacian_matrix()
True
"""
E = Gamma.edges()
V = Gamma.vertices()
IG = [[incidence_value(Gamma, v, e, eo) for v in V] for e in E]
return matrix(QQ, IG).transpose()
> Vincent
>
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