Suppose we have an adjacency matrix A representation of a directed graph G.
Which of the following must be true for any adjacency matrix, regardless of the underlying graph?Some matrix definitions:
\(A^T\) denotes the transpose of a matrix \(A\), such that if \(A\) has \(m\) rows and \(n\) columns, \(A^T\) has \(n\) and \(m\) columns. Additionally, the value of the cell \((i, j)\) in \(A\) becomes the value of the cell \((j, i)\) in \(A^T\).
There is a length 2 path iff there exists some k such that \(i, k, j\) are connected iff exists index k such that \((i, k) = 1\) and \((k, j) = 1\) iff row \(i\) times column \(j\) is not 0 iff In \(A^2, (i, j)\) is not zero.
If two rows in an adjacency matrix are the same, then this just represents that two vertices have edges leading to the same set of vertices.