# 2.5 Digraphs: Walks & Paths

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 $$A^T$$.

Exercise 1

For an undirected graph, the transpose of a matrix is itself because having an edge from $$i$$ to $$j$$ is equivalent to having an edge from $$i$$ to $$j$$. However, for directed graphs, this is not the case because these two edges would point in opposite directions.
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.