Meaning of the adjacency matrix product
This is answered on  Math.SE; see also  MathWorld . 

Briefly,the $(i,j)$-th entry in $A^n$ gives the number of directed walks from vertex $i$ to $j$ that have length $n$. Also, given another adjacency matrix $B$, this entry in the product $(AB)$ gives the number of directed walks from vertex $i$ to $j$ , that walk first along an edge of the first graph and then along an edge of the second.Putting these two together would give you an interpretation to the product of 7th powers in your question.