Exploring the dimensions

This adjacency matrix represents a cube. There are 6 unique paths that have a distance of 2 from any given node on this figure, however these paths can be paired up to trace out the 1 face of the cube, and thus both go to the same node. Thus there are 3 nodes that have a distance of two from any given node.

There are 8 nodes, so there are 8*3=24 such pairs.

Note: You will have the each pair of nodes represented twice: once as (a,b) and once as (b,a). So this problem is a little ambiguous in that it doesn't say ordered pairs of points, thus 12 could be argued as a correct answer as well.

If $A=(a_{ij})$ is the adjacency matrix, compute $B=(b_{ij})=A^2$ , which represents the number of paths of length 2 from vertex $x_i$ to vertex $x_j$ .

For each $1\leq i\neq j\leq 8$ , count 1 if $a_{ij}=0$ and $b_{ij}>0$ .