(Solved): 1. (a) Consider the directed graph shown below, in which the vertices have been numbered according ...

1. (a) Consider the directed graph shown below, in which the vertices have been numbered according to a certain labelling \( \lambda \). i. [5] Give its adjacency matrix \( M^{\lambda} \) and use it to calculate the number of paths of length four which start at vertex-2 and end at vertex-3. ii. [15] Let \( \mu \) be a labelling of the same graph such that \( \lambda(1)=\mu(2) \), \( \lambda(2)=\mu(3) \) and \( \lambda(3)=\mu(1) \). Give a representation of the graph with labelling \( \mu \), and find the adjacency matrix \( M^{\mu} \). Find the similarity transformation which relates \( M^{\mu} \) and \( M^{\lambda} \) and show how \( M^{\mu} \) is obtained from \( M^{\lambda} \).