### Defining the matrices for various rings
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def distLapla(G):
n=G.order()
M=zero_matrix(RDF,n,n)
for i in range(n):
for j in range(i+1,n):
x=G.distance(i,j)
M[i,j]=-x
M[j,i]=-x
M[i,i]=M[i,i]+x
M[j,j]=M[j,j]+x
return M
def InvTransMatrix(G):
n=G.order()
M=zero_matrix(RDF,n,n)
for i in range(n):
for j in range(i+1,n):
x=G.distance(i,j)
M[i,j]=0
M[j,i]=0
M[i,i]=M[i,i]+x
M[j,j]=M[j,j]+x
for i in range(n):
M[i,i]=M[i,i]^(-1)
return M
def InvTransDistLapla(G):
M=InvTransMatrix(G)*distLapla(G)
return M
def distLaplaQQ(G):
n=G.order()
M=zero_matrix(QQ,n,n)
for i in range(n):
for j in range(i+1,n):
x=G.distance(i,j)
M[i,j]=-x
M[j,i]=-x
M[i,i]=M[i,i]+x
M[j,j]=M[j,j]+x
return M
def InvTransMatrixQQ(G):
n=G.order()
M=zero_matrix(QQ,n,n)
for i in range(n):
for j in range(i+1,n):
x=G.distance(i,j)
M[i,j]=0
M[j,i]=0
M[i,i]=M[i,i]+x
M[j,j]=M[j,j]+x
for i in range(n):
M[i,i]=M[i,i]^(-1)
return M
def InvTransDistLaplaQQ(G):
M=InvTransMatrixQQ(G)*distLaplaQQ(G)
return M
def distLaplaSR(G):
n=G.order()
M=zero_matrix(SR,n,n)
for i in range(n):
for j in range(i+1,n):
x=G.distance(i,j)
M[i,j]=-x
M[j,i]=-x
M[i,i]=M[i,i]+x
M[j,j]=M[j,j]+x
return M
def HalfInvTransMatrixSR(G):
n=G.order()
M=zero_matrix(SR,n,n)
for i in range(n):
for j in range(i+1,n):
x=G.distance(i,j)
M[i,j]=0
M[j,i]=0
M[i,i]=M[i,i]+x
M[j,j]=M[j,j]+x
for i in range(n):
M[i,i]=M[i,i]^(-1/2)
return M
def NormDistLaplaSR(G):
M=HalfInvTransMatrixSR(G)*distLaplaSR(G)*HalfInvTransMatrixSR(G)
return M
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### Using the built in cospectral_graphs command to search for <=8 vertices
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L=[]
for G in graphs.cospectral_graphs(3, matrix_function=InvTransDistLaplaQQ, graphs=lambda g: g.is_connected()):
L.append([i.graph6_string() for i in G])
L
[] [] |
L=[]
for G in graphs.cospectral_graphs(4, matrix_function=InvTransDistLaplaQQ, graphs=lambda g: g.is_connected()):
L.append([i.graph6_string() for i in G])
L
[] [] |
L=[]
for G in graphs.cospectral_graphs(5, matrix_function=InvTransDistLaplaQQ, graphs=lambda g: g.is_connected()):
L.append([i.graph6_string() for i in G])
L
[] [] |
L=[]
for G in graphs.cospectral_graphs(6, matrix_function=InvTransDistLaplaQQ, graphs=lambda g: g.is_connected()):
L.append([i.graph6_string() for i in G])
L
[] [] |
L=[]
for G in graphs.cospectral_graphs(7, matrix_function=InvTransDistLaplaQQ, graphs=lambda g: g.is_connected()):
L.append([i.graph6_string() for i in G])
L
[] [] |
L=[]
for G in graphs.cospectral_graphs(8, matrix_function=InvTransDistLaplaQQ, graphs=lambda g: g.is_connected()):
L.append([i.graph6_string() for i in G])
L
[['GfC}@o', 'GraK[S']] [['GfC}@o', 'GraK[S']] |
G1=Graph('GfC}@o')
G2=Graph('GraK[S')
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show(G1)
NDL1=NormDistLaplaSR(G1)
show(NDL1)
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show(G2)
NDL2=NormDistLaplaSR(G2)
show(NDL2)
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InvTransDistLaplaQQ(G1).charpoly()-InvTransDistLaplaQQ(G2).charpoly()
0 0 |
print InvTransDistLaplaQQ(G1).eigenvalues()
print InvTransDistLaplaQQ(G1).charpoly()
[12/11, 1, 0, 7/6, 7/6, 1.039501700164266?, 1.113380330521739?, 1.422875545071572?] x^8 - 8*x^7 + 317947/11616*x^6 - 5428399/104544*x^5 + 24668087/418176*x^4 - 4196075/104544*x^3 + 575771/38016*x^2 - 85211/34848*x [12/11, 1, 0, 7/6, 7/6, 1.039501700164266?, 1.113380330521739?, 1.422875545071572?] x^8 - 8*x^7 + 317947/11616*x^6 - 5428399/104544*x^5 + 24668087/418176*x^4 - 4196075/104544*x^3 + 575771/38016*x^2 - 85211/34848*x |