Category:3-bit Walsh permutations; neighbor graph
Each of the 168 invertible binary 3×3 matrices has between 2 and 6 neighbors, to which it differs in only one entry.
This neighbor relationship forms a graph with two components, each with 84 vertices.
One contains the matrices with real determinant 1, the other one those with real determinant −1.
(This can be observed interactively by clicking on the links in the matrices in categories like Related images of Walsh permutation 124.)
Positive component[edit]
The connected component is separable.
There are 3 clusters with 25 vertices each. (They are biconnected components.)
Between any pair of clusters are links, consisting of a link node and two edges. (So the graph is triconnected.)
The clusters can be seen as a grid of 6 octants, with 2 opposite ones missing.
The cluster nodes connected to the link nodes are the vertices of the Petrie hexagons (drawn with thicker edges).
(The negative equivalents to these images are just their reflections. E.g. this is the negative component, and this is the negative link matrix.)
The coordinate systems under each cluster show which changing matrix fields correspond to the 8 edges in the respective direction.
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Conjugacy classes[edit]
The node colors in these images denote the 6 conjugacy classes. Those in the two components are completely different.
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Complement patterns[edit]
The vertex colors in these images denote complement patterns, which are numbers 1..7. (They correspond to rows 1..7 in a bitwise XOR matrix.)
Vertices in reflected positions have reflected binary numbers: The reflection of 1 (red) is 4 (blue), and the reflection of 3 (yellow) is 6 (cyan).
2 (green), 5 (magenta) and 7 (white) are not changed by reflection.
There are 12 nodes of each color in each image. Of the red, green and blue nodes there are 3 in each cluster and 3 between. Of all others there are 4 in each cluster.
The red, green and blue nodes form a complete cycle for each color.
Subcategories
This category has the following 3 subcategories, out of 3 total.
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Pages in category "3-bit Walsh permutations; neighbor graph"
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