# Meaning of the adjacency matrix product

Let $A$ be an adjacency matrix of a directed graph. What's the meaning of the $(i,j)-$entry of the matrix $((A^T)^{7} \cdot (A^{7}))$ ?

My initial interpretation is that $(i,j)$ of this matrix is zero whenever nodes $i$ and $j$ have no 7-length in-coming paths from a common source. Is that right? Any attention is appreciated!

• Have you tried calculating the thing on a small example? Why are you interested in that quantity?
– Raphael
Oct 7, 2012 at 22:36

## 1 Answer

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.

• Since $B=A^T$ here, there might be more/different meaning.
– Raphael
Oct 7, 2012 at 22:38
• My initial interpretation is that $(i,j)$ of this matrix is zero whenever nodes $i$ and $j$ have no 7-length in-coming paths from a common source. Is that right? Oct 8, 2012 at 3:11