Consider the following graph.

a$)$ I$)$ Write down the adjacency matrix for the related undirected

graph $($i$.$e$.$ ignoring all directional arrowheads and regarding

all edges as undirected.$)$

ii$)$ By squaring this matrix, calculate how many paths of length

$2$ exist starting at Node $5$ and ending at Node $2.$

iii$)$ Write down all paths of length $2$ which start at Node $5$ and

ending at Node $2$ and hence verify the result from part ii$).$

Note: A path containing a loop will be double counted.

b$)$ I$)$ Write down the adjacency matrix for the directed graph as

shown, taking the direction of each edge into account.

ii$)$ By squaring this matrix, calculate how many paths of length

$2$ exist starting at Node $5$ and ending at Node $2.$

iii$)$ Write down all paths of length $2$ which start at Node $5$ and

ending at Node $2$ and hence verify the result from part ii$).$