Find which town is last to be polluted. Justify your answer.

Write down the number of days it takes for the pollution to reach the last town.

Write down the adjacency matrix for this network.

Determine the number of different walks of length $5$ that start and end at the same vertex.

$q$.

$p$.

Use the nearest neighbour algorithm to find two possible cycles.

Hence state the value of the upper bound for Daniel’s travel time for the value of $x$ found in part (e)(i).

Use Prim’s algorithm, starting at vertex $\text{B}$, to find the weight of the minimum spanning tree of the remaining graph. You should indicate clearly the order in which the algorithm selects each edge.

Find the least value of $x$ for which the edge $\text{AF}$ will definitely not be used by Daniel.

Hence, for the case where $x<9$, find a lower bound for Daniel’s travel time, in terms of $x$.

Find the number of distinct walks of length 4 beginning and ending at A.

Write down the adjacency matrix for G.

Find the number of ways that the ant can start at the vertex $\text{A}$, and walk along exactly $6$ edges to return to $\text{A}$.

$r$.

Find the best upper bound for Daniel’s travel time.

Write down a Hamiltonian cycle in $W$.

The tourist office in the city has received complaints about the lack of cleanliness of some routes between the attractions. Corinne, the office manager, decides to inspect all the routes between all the attractions, starting and finishing at $\text{H}$. The sum of the weights of all the edges in graph $W$ is $(92+x)$.

Corinne inspects all the routes as quickly as possible and takes $2$ hours.

Find the value of $x$ during Corinne’s inspection.