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Eigenfunctions of the Graph Laplacian

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Eigenfunctions of the Graph Laplacian

Each PDF file below contains a list of all eigenvalues along with pictures of the first eigenfunctions. For non-glued identifications, Dirichlet boundaries were used. Soon it will also be possible to obtain the numerical values for eigenvalues and eigenfunctions as Python lists.

Eigenfunction Approximations

Identification Type	Level 1	Level 2	Level 3	Level 4 (First 150)
Torus				PDF
Torus Glued	PDF	PDF	PDF	PDF
Projective Plane				PDF
Projective Plane Glued	PDF	PDF	PDF	PDF
Klein Bottle (Horizontal)				PDF
Kelin Bottle (Horizontal) Glued	PDF	PDF	PDF	PDF

Identification Type	Level 4 (First 15)
Ident(T,T,T,T,T)	First 15
Ident(P,P,P,P,T)	First 15
Ident(T,P,T,P,T)	First 15
Ident(P,T,P,T,T)	First 15
Ident(KH,KV,KH,KV,T)	First 15
Ident(T,KH,T,KH,T)	First 15
Ident(KH,T,KH,T,T)	First 15
Ident(P,KH,P,KH,T)	First 15
Ident(KH,P,KH,P,T)	First 15
Ident(T,P,P,P,T)	First 15
Ident(P,T,P,P,T)	First 15
Ident(P,P,T,P,T)	First 15
Ident(P,P,P,T,T)	First 15
Ident(P,T,T,T,T)	First 15
Ident(T,P,T,T,T)	First 15
Ident(T,T,P,T,T)	First 15
Ident(T,T,T,P,T)	First 15
Ident(T,KH,KH,KH,T)	First 15
Ident(KH,T,KH,KH,T)	First 15
Ident(KH,KH,T,KH,T)	First 15
Ident(KH,KH,KH,T,T)	First 15
Ident(KH,T,T,T,T)	First 15
Ident(T,KH,T,T,T)	First 15
Ident(T,T,KH,T,T)	First 15
Ident(T,T,T,KH,T)	First 15
Ident(P,KH,KH,KH,T)	First 15
Ident(KH,P,KH,KH,T)	First 15
Ident(KH,KH,P,KH,T)	First 15
Ident(KH,KH,KH,P,T)	First 15
Ident(KH,P,P,P,T)	First 15
Ident(P,KH,P,P,T)	First 15
Ident(P,P,KH,P,T)	First 15
Ident(P,P,P,KH,T)	First 15
Ident(T,P,KH,T,T)	First 15
Ident(P,KH,T,P,T)	First 15
Ident(KH,T,P,KH,T)	First 15
Ident(P,P,T,T,T)	First 15
Ident(T,T,P,P,T)	First 15
Ident(P,T,P,T,T)	First 15
Ident(T,P,T,P,T)	First 15

Comparison of Eigenfunctions by Averaging

The files below refer to a "dot value" which in general measures the matching quality. Dot values are in general between 0 and 1:

• those close to 0 are better matches, while
• those close to 1 are not good matches.

Dot values above 1 are somewhat artificial markers:

• dot value 2 indicates the eigenfunction averages to the zero function, and
• dot value 3 indicates the projection onto the closest eigenspace is zero.

A full description of the dot value may be seen here (PDF).

Identification Type	Level 2 to 1	Level 3 to 2	Level 4 to 3
Torus Glued	Pictures (PDF) Matched Eigenvalue Values (PDF) Matched Eigenvalue Indices (PDF) Dot Values (txt)	First 150 Pictures (PDF) Matched Eigenvalue Values (PDF) Matched Eigenvalue Indices (PDF) Dot Values (txt)	First 150 Pictures (PDF) Matched Eigenvalue Values (PDF) Matched Eigenvalue Indices (PDF) Dot Values (txt)
Projective Plane Glued	Pictures (PDF) Matched Eigenvalue Values (PDF) Matched Eigenvalue Indices (PDF) Dot Values (txt)	First 150 Pictures (PDF) Matched Eigenvalue Values (PDF) Matched Eigenvalue Indices (PDF) Dot Values (txt)	First 150 Pictures (PDF) Matched Eigenvalue Values (PDF) Matched Eigenvalue Indices (PDF) Dot Values (txt)
Kelin Bottle (Horizontal) Glued	Pictures (PDF) Matched Eigenvalue Values (PDF) Matched Eigenvalue Indices (PDF) Dot Values (txt)	First 150 Pictures (PDF) Matched Eigenvalue Values (PDF) Matched Eigenvalue Indices (PDF) Dot Values (txt)	First 150 Pictures (PDF) Matched Eigenvalue Values (PDF) Matched Eigenvalue Indices (PDF) Dot Values (txt)

Eigenvalue Ratios

Eigenvalues of multiplicity are marked with asterisks.

Identification Type	Eigenvalue Ratios
Torus Glued	PDF
Projective Plane Glued	PDF
Klein Bottle Horizontal Glued	PDF

Eigenvalues

Identification Type	Eigenvalues (Python List)
Ident(KH,KH,KH,KH,T)	Eigenvalue List
Ident(KH,KH,KH,KV,T)	Eigenvalue List
Ident(KH,KH,KH,P,T)	Eigenvalue List
Ident(KH,KH,KH,T,T)	Eigenvalue List
Ident(KH,KH,KV,KH,T)	Eigenvalue List
Ident(KH,KH,KV,KV,T)	Eigenvalue List
Ident(KH,KH,KV,P,T)	Eigenvalue List
Ident(KH,KH,KV,T,T)	Eigenvalue List
Ident(KH,KH,P,KH,T)	Eigenvalue List
Ident(KH,KH,P,KV,T)	Eigenvalue List
Ident(KH,KH,P,P,T)	Eigenvalue List
Ident(KH,KH,P,T,T)	Eigenvalue List
Ident(KH,KH,T,KH,T)	Eigenvalue List
Ident(KH,KH,T,KV,T)	Eigenvalue List
Ident(KH,KH,T,P,T)	Eigenvalue List
Ident(KH,KH,T,T,T)	Eigenvalue List
Ident(KH,KV,KH,KH,T)	Eigenvalue List
Ident(KH,KV,KH,KV,T)	Eigenvalue List
Ident(KH,KV,KH,P,T)	Eigenvalue List
Ident(KH,KV,KH,T,T)	Eigenvalue List
Ident(KH,KV,KV,KH,T)	Eigenvalue List
Ident(KH,KV,KV,KV,T)	Eigenvalue List
Ident(KH,KV,KV,P,T)	Eigenvalue List
Ident(KH,KV,KV,T,T)	Eigenvalue List
Ident(KH,KV,P,KH,T)	Eigenvalue List
Ident(KH,KV,P,KV,T)	Eigenvalue List
Ident(KH,KV,P,P,T)	Eigenvalue List
Ident(KH,KV,P,T,T)	Eigenvalue List
Ident(KH,KV,T,KH,T)	Eigenvalue List
Ident(KH,KV,T,KV,T)	Eigenvalue List
Ident(KH,KV,T,P,T)	Eigenvalue List
Ident(KH,KV,T,T,T)	Eigenvalue List
Ident(KH,P,KH,KH,T)	Eigenvalue List
Ident(KH,P,KH,KV,T)	Eigenvalue List
Ident(KH,P,KH,P,T)	Eigenvalue List
Ident(KH,P,KH,T,T)	Eigenvalue List
Ident(KH,P,KV,KH,T)	Eigenvalue List
Ident(KH,P,KV,KV,T)	Eigenvalue List
Ident(KH,P,KV,P,T)	Eigenvalue List
Ident(KH,P,KV,T,T)	Eigenvalue List
Ident(KH,P,P,KH,T)	Eigenvalue List
Ident(KH,P,P,KV,T)	Eigenvalue List
Ident(KH,P,P,P,T)	Eigenvalue List
Ident(KH,P,P,T,T)	Eigenvalue List
Ident(KH,P,T,KH,T)	Eigenvalue List
Ident(KH,P,T,KV,T)	Eigenvalue List
Ident(KH,P,T,P,T)	Eigenvalue List
Ident(KH,P,T,T,T)	Eigenvalue List
Ident(KH,T,KH,KH,T)	Eigenvalue List
Ident(KH,T,KH,KV,T)	Eigenvalue List
Ident(KH,T,KH,P,T)	Eigenvalue List
Ident(KH,T,KH,T,T)	Eigenvalue List
Ident(KH,T,KV,KH,T)	Eigenvalue List
Ident(KH,T,KV,KV,T)	Eigenvalue List
Ident(KH,T,KV,P,T)	Eigenvalue List
Ident(KH,T,KV,T,T)	Eigenvalue List
Ident(KH,T,P,KH,T)	Eigenvalue List
Ident(KH,T,P,KV,T)	Eigenvalue List
Ident(KH,T,P,P,T)	Eigenvalue List
Ident(KH,T,P,T,T)	Eigenvalue List
Ident(KH,T,T,KH,T)	Eigenvalue List
Ident(KH,T,T,KV,T)	Eigenvalue List
Ident(KH,T,T,P,T)	Eigenvalue List
Ident(KH,T,T,T,T)	Eigenvalue List
Ident(P,KH,KH,KH,T)	Eigenvalue List
Ident(P,KH,KH,KV,T)	Eigenvalue List
Ident(P,KH,KH,P,T)	Eigenvalue List
Ident(P,KH,KH,T,T)	Eigenvalue List
Ident(P,KH,KV,KH,T)	Eigenvalue List
Ident(P,KH,KV,KV,T)	Eigenvalue List
Ident(P,KH,KV,P,T)	Eigenvalue List
Ident(P,KH,KV,T,T)	Eigenvalue List
Ident(P,KH,P,KH,T)	Eigenvalue List
Ident(P,KH,P,KV,T)	Eigenvalue List
Ident(P,KH,P,P,T)	Eigenvalue List
Ident(P,KH,P,T,T)	Eigenvalue List
Ident(P,KH,T,KH,T)	Eigenvalue List
Ident(P,KH,T,KV,T)	Eigenvalue List
Ident(P,KH,T,P,T)	Eigenvalue List
Ident(P,KH,T,T,T)	Eigenvalue List
Ident(P,P,KH,KH,T)	Eigenvalue List
Ident(P,P,KH,KV,T)	Eigenvalue List
Ident(P,P,KH,P,T)	Eigenvalue List
Ident(P,P,KH,T,T)	Eigenvalue List
Ident(P,P,P,KH,T)	Eigenvalue List
Ident(P,P,P,P,T)	Eigenvalue List
Ident(P,P,P,T,T)	Eigenvalue List
Ident(P,P,T,KH,T)	Eigenvalue List
Ident(P,P,T,P,T)	Eigenvalue List
Ident(P,P,T,T,T)	Eigenvalue List
Ident(P,T,KH,KH,T)	Eigenvalue List
Ident(P,T,KH,KV,T)	Eigenvalue List
Ident(P,T,KH,P,T)	Eigenvalue List
Ident(P,T,KH,T,T)	Eigenvalue List
Ident(P,T,P,KH,T)	Eigenvalue List
Ident(P,T,P,P,T)	Eigenvalue List
Ident(P,T,P,T,T)	Eigenvalue List
Ident(P,T,T,KH,T)	Eigenvalue List
Ident(P,T,T,P,T)	Eigenvalue List
Ident(P,T,T,T,T)	Eigenvalue List
Ident(T,KH,KH,KH,T)	Eigenvalue List
Ident(T,KH,KH,KV,T)	Eigenvalue List
Ident(T,KH,KH,P,T)	Eigenvalue List
Ident(T,KH,KH,T,T)	Eigenvalue List
Ident(T,KH,KV,KH,T)	Eigenvalue List
Ident(T,KH,KV,KV,T)	Eigenvalue List
Ident(T,KH,KV,P,T)	Eigenvalue List
Ident(T,KH,KV,T,T)	Eigenvalue List
Ident(T,KH,P,KH,T)	Eigenvalue List
Ident(T,KH,P,KV,T)	Eigenvalue List
Ident(T,KH,P,P,T)	Eigenvalue List
Ident(T,KH,P,T,T)	Eigenvalue List
Ident(T,KH,T,KH,T)	Eigenvalue List
Ident(T,KH,T,KV,T)	Eigenvalue List
Ident(T,KH,T,P,T)	Eigenvalue List
Ident(T,KH,T,T,T)	Eigenvalue List
Ident(T,P,KH,KH,T)	Eigenvalue List
Ident(T,P,KH,KV,T)	Eigenvalue List
Ident(T,P,KH,P,T)	Eigenvalue List
Ident(T,P,KH,T,T)	Eigenvalue List
Ident(T,P,P,KH,T)	Eigenvalue List
Ident(T,P,P,P,T)	Eigenvalue List
Ident(T,P,P,T,T)	Eigenvalue List
Ident(T,P,T,KH,T)	Eigenvalue List
Ident(T,P,T,P,T)	Eigenvalue List
Ident(T,P,T,T,T)	Eigenvalue List
Ident(T,T,KH,KH,T)	Eigenvalue List
Ident(T,T,KH,KV,T)	Eigenvalue List
Ident(T,T,KH,P,T)	Eigenvalue List
Ident(T,T,KH,T,T)	Eigenvalue List
Ident(T,T,P,KH,T)	Eigenvalue List
Ident(T,T,P,P,T)	Eigenvalue List
Ident(T,T,P,T,T)	Eigenvalue List
Ident(T,T,T,KH,T)	Eigenvalue List
Ident(T,T,T,P,T)	Eigenvalue List
Ident(T,T,T,T,T)	Eigenvalue List