# 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!

## migrated from cstheory.stackexchange.comOct 7 '12 at 16:28

This question came from our site for theoretical computer scientists and researchers in related fields.

• Have you tried calculating the thing on a small example? Why are you interested in that quantity? – Raphael Oct 7 '12 at 22:36

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 '12 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? – John Smith Oct 8 '12 at 3:11