Are you speaking about the set of vertices
$$\bigcup_{i=0}^7\bigcup_{j=1}^7\{(i,j,0),(j,0,i),(0,i,j)\}$$ i.e. the grid points on three faces except the corner
$(0,0,0)$? Also, are two vertices adjacent if they are close in Euclidean space, i.e.
$|a-b|\leq1$? Assuming these, I jump to
With[{verts=Flatten[Table[{{i,j,0},{j,0,i},{0,i,j}},{i,0,7},{j,1,7}],2]},
AdjacencyMatrix@Graph[UndirectedEdge@@@Select[
Subsets[verts,{2}],#[[1]]!=#[[2]]\[And]Norm[#[[1]]-#[[2]]]<=1&
]]
]
which is very inefficient but gets the job done. I hesitate to attempt a MatrixForm on that chunker. It's definitely possible to exploit meshes and built in geomtry to get this data "for free", but specifying it isn't too bad especially if we lazily construct edges out of all pairs of vertices.
