Surface of the 3x3x3 cube as a graph
APL (Dyalog Classic), 34 30 bytes

-4 thanks to jimmy23013



4≥+/¨|∘.-⍨,(⍳3)∘.⌽7 ¯1∘.,○⍳3 3


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outputs an adjacency matrix with each vertex adjacent to itself

⍳3 3 generate an array of (0 0)(0 1)(0 2)(1 0)(1 1)(1 2)(2 0)(2 1)(2 2)

○ multiply all by π

7 ¯1∘., prepend 7 or -1 in all possible ways

(⍳3)∘.⌽ rotate coord triples by 0 1 2 steps in all possible ways

+/¨|∘.-⍨, compute manhattan distance between each pair

4≥ it must be no greater than 4 for neighbouring facets