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Meaning of the adjacency matrix product
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Problem
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!
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!
Solution
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.
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.
Context
StackExchange Computer Science Q#4923, answer score: 7
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