projectiv_plane.ipynb

Kernel: SageMath 8.1

M = designs.DesarguesianProjectivePlaneDesign(9) #(91, 10, 1) design
B = M.blocks() #len(B) is 91
varieties = []

#collect all elements into varieties
for i in range(91):
    varieties += [elm for elm in B[i]]
varieties = list(set(varieties)) #Store all elements
varieties.sort() #not important

#map (a, b, c) --> n a natural number 
dic = {varieties[i]:i for i in range(len(varieties))}
num = lambda x: dic[x] #return the numerical value
B_new = [map(num,B[i]) for i in range(len(B))] #Each block has numerical entries which are easy to handle for indexing

#This function returns 1 if an element i in the block j, 0 otherwise 
yesNo = lambda i, j, matrix: 1 if i in matrix[j] else 0 #Following the definition of the incidence matrix
#Given blocks we would like to construct its incidence matrix
incMat = lambda Mdesig: Matrix([[yesNo(i, j, Mdesig) for i in range(len(Mdesig))] for j in range(len(Mdesig))])#Return

def graph_mat(matrix):
    """Returns the edges of a graph given its matrix"""
    length = matrix.dimensions()[0] 
    Block = []
    for i in range(length):
        for j in range(length):
            if matrix[i][j] == 1:
                Block.append((i,j))
    Block.sort() #This is more complicated but it doesn't have a bug!
    return Block


####Experiment####
matrix = incMat(B_new) #incidence matrix
G = Graph(graph_mat(matrix))
G_T = Graph(graph_mat(matrix.transpose()))
print( G.is_isomorphic(G_T)) #True unfortunately </code>

/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/repl/ipython_kernel/__main__.py:35: DeprecationWarning: You created a graph with multiple edges from a list. Please set 'multiedges' to 'True' when you do so, as in the future the default behaviour will be to ignore those edges
See http://trac.sagemath.org/15706 for details.
/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/repl/ipython_kernel/__main__.py:35: DeprecationWarning: You created a graph with loops from a list. Please set 'loops' to 'True' when you do so, as in the future the default behaviour will be to ignore those edges
See http://trac.sagemath.org/15706 for details.
/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/repl/ipython_kernel/__main__.py:36: DeprecationWarning: You created a graph with multiple edges from a list. Please set 'multiedges' to 'True' when you do so, as in the future the default behaviour will be to ignore those edges
See http://trac.sagemath.org/15706 for details.
/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/repl/ipython_kernel/__main__.py:36: DeprecationWarning: You created a graph with loops from a list. Please set 'loops' to 'True' when you do so, as in the future the default behaviour will be to ignore those edges
See http://trac.sagemath.org/15706 for details.

True