George, an experienced civiby.i 2 -g4mq qw(aui3zl engineer, is designing a high-speed railway network to connect seven cities (A-G), as illustrated in the graph below. The possible railways and costs of building them are represented by the graph edges, where costs am .( i zi2q-uy3qbgw4are in billions of dollars.


George needs to design the lowest cost high-speed railway network that will connect the seven cities. Each city does not need to directly connect with all other cities, however each city must have a connection to the rail network.
1.Name an algorithm which will allow George to find the lowest cost high-speed railway network.
2.(1)Find and list the railway connections of the lowest cost high-speed railway network.
(2)Hence, write down the cost of building this network.

A local government is plann6iyvc j yng8sxq4:i v:v3jg/9ing to connect nine nearby mountain villages (A-I) to a new 9cj4s qi83gji:vv:g/ n 6yvy xlocal airport (O) with roads, as illustrated on the graph below.

The numbers on the graph edges represent the distance in kilometres between the villages and airport.


Every village does not need to be connected directly, however it must be possible to traverse to the airport from any village.
1.Determine the most efficient way to connect the villages to the airport.
2.Determine the total length of road the local government will need to build.

Omni Communications Inc. is planning to connect )uyil: gtg( tco v49r(six cities (A-F) with a fibre optic network. Due to the high speed of information tgy(u(t igto 4vl9) cr:ransfer over a fibre optic network, each city does not need to be connected directly to each other.
The numbers on the graph below indicate the cost, in thousands of dollars, for each possible connection between the cities.



Determine the network of cables that will minimize the total cost of connecting these six cities to the fibre optic network, and the associated cost.

  Peter is a salesman who works for a skin care products company. q7jd u y)9b6p3i)zpfv,ksee 0 He travels to a nearby city to find new ep)6kipsfdz,qj) ebvuy30 7 9customers. He researches six beauty salons on Yelp and pins their locations on Google Map.

The numbers on the graph below indicate the travel times in minutes between Peter's hotel (at A) and the beauty salons (B-G).

Peter is currently located at his hotel and wants to visit each beauty salon once, before returning to the hotel.


1.Write down the term which is used to describe this type of route.

Peter decides to visit beauty salon E first.
2.Find the route that he can take.

3.Calculate the total travel time, in hours and minutes, for his route.    hours    minutes

A biotechnology company is investigating the costs of buildingteu+9: 8xog rl a fibre optic network to connect the 8 research labs it owns. The weighted graph below shows the labs (A-H) and the cost, in thousands of dollars, for b:xr e+9l8g utouilding each connection.
Due to the fast information flow provided by fibre optic networks, the company decides that each lab does not need to be directly connected to each other, however each lab needs to have a connection to the overall network.

1.(1)Using Kruskal's algorithm, find a minimum spanning tree for the graph. Clearly indicate the order in which the edges are added to the tree.
(2)Hence draw the minimum spanning tree for the graph.
2.Find the minimum cost of building the fibre optic network.

  A team of network technicians at a cloud storage company randomly move betwnr-s0xu/m- 0icn8wqxgcz , b 0een data centers A, B, C a0-s /8n g,b qucx00z wircmxn-nd D, with probabilities shown on the graph below.

1.Find the value of p and the value of q. p =    q =   
2.(1)Complete the weighted adjacency table below.

(2)Hence write down a transition matrix A for the graph.
3.Given that the team is currently in data center C, find the probability that they will return to C after visiting exactly two other data centers.

A postman starts at the Post Office located at O and needs to tra40um-2aj hm+wo hl 9ixvel along every road +jih4-9 w mam2lxhou0 illustrated by the edges in the graph below. The numbers on the graph represent distances in kilometres.

1.Find a solution to the Chinese Postman Problem for the graph.
2.Write down the length, in kilometres, of an optimal Chinese postman route.

A package delivery company offers 24 hours delivery service to a d.8ecks*2bzvait5 3j limited number of cities in close proximity. The directed graph below shows four cities served (A-D) and the routes of deliveryied ck*3 j8a 2tbv5s.z between them.



1. Find an adjacency matrix $\boldsymbol{A}$ for the shipping network.
2. Find the number of ways it is possible to ship a package from city $\mathrm{C}$ to city $\mathrm{A}$ using exactly three delivery routes.
3. (1) Find the matrix $\boldsymbol{S}_{3}$ , where $\boldsymbol{S}_{3}=\sum_{i=1}^{3} \boldsymbol{A}^{i}$ .
(2) Hence write down the number of ways it is possible to deliver a package from city A to city D using three or less delivery routes.
(3)State all possible ways to ship a package from city A to city D using three or less delivery routes.

The graph $\boldsymbol{G}$ as the adjacency matrix $\boldsymbol{M}$given below.



1. Draw a possible graph of $\boldsymbol{G}$ .
2. Interpret the meaning of the diagonal elements of $\boldsymbol{M}^{2}$ , with regards to $boldsymbol{G}$ .
3. Find the number of walks of length 4 starting at A and ending at D .
4. List the two trails of length 4 starting at A and ending at D .

The tourism and travel company, Zeppztns0harv/0n 0 a6*tu8j , 7koujnef 8elin Adventure, provides airship trips between five cities of Spain. The airship trip routes are shown on the map8a0efn0tk6j*trv,8u/n ajn0u zo 7sh below.



1. Write down the number of trip routes offered.
2. Find an adjacency matrix $\boldsymbol{A}$ for the airship trips network.
3. Find:
(1) the number of routes, m , between two cities that cannot be made using at most one stopover.
(2) the maximum number of trips, k , needed to fly between any two cities.

Zeppelin Adventure is considering offering a new trip option for customers. They are considering either a new route from Barcelona to Valencia, or make the route between Madrid and Zaragoza two -way.
4. Determine the route that minimises the value of m , from part (c) (i).

A postman starts at the Post Office located at O and needs to travel along ev1-n5 dw-ky ccqery road illustrated by the edges in the graph below. The numbers on the graph represent distances in kilom1w-dy5-ccknq etres of the roads.



1. Find a solution to the Chinese Postman Problem for the graph.
2. Write down the length, in kilometres, of an optimal Chinese postman route.

The complete graph $\boldsymbol{K}$ has the following weighted adjacency table.



Consider the travelling salesman problem for $\boldsymbol{K}$ .
1. By first finding a minimum spanning tree on the subgraph of $\boldsymbol{K}$ formed by deleting vertex $\mathrm{A}$ and all edges connected to $\mathrm{A}$ , find a lower bound to this problem.
2. Find the total weight of the cycle ABCEDA.
3. State a conclusion from your results found in part (a) and part (b).

The weights of the edges of wj;y 25 :bjjht1iyw t7a graph $\boldsymbol{H}$ are given in the following table.


1. (1) Draw the weighted graph $\boldsymbol{H}$ on the vertices below.


(2) Using Kruskal's algorithm, find a minimum spanning tree for $\boldsymbol{H}$ . Clearly indicate the order in which the edges are added to the tree.
(3) Write down the weight of the minimum spanning tree.

Consider the following weighted graph $\boldsymbol{J}$ .



2. (1) Find a solution to the Chinese postman problem for the graph $\boldsymbol{J}$ .
(2) Write down the total weight of the solution.
3. (1) State the travelling salesman problem.
(2) Explain why there is no solution to the travelling salesman problem for the graph $\boldsymbol{J}$ .

The weighted graph $\boldsymbol{H}$, representing the travelling costs between five customers, has the following weighted adjacency table.


1. Draw a possible graph of $\boldsymbol{H}$ .
2. Starting from customer $\mathrm{E}$ , use the nearest-neighbour algorithm to determine an upper bound to the travelling salesman problem for $\boldsymbol{H}$ .
3. By removing customer A, use the method of vertex deletion to determine a lower bound to the travelling salesman problem for $\boldsymbol{H}$ .

A baker, starting a business baking bread and muffins, pl5 74c7i 4mkhnppgk* cx lc84onans to visit 5 nearby cafes (A-E) to enquire if they want to stock her goods in their cafes. The distances between the cafes, in km, are shown in the fpc 7inox nhk 4*845l7cmg4k cpollowing weighted adjacency table.

1.Draw a possible graph for this weighted adjacency table.
2.Starting from cafe C, use the nearest-neighbour algorithm to determine an upper bound to the travelling salesman problem for this graph.
3.By removing cafe E, use the method of vertex deletion to determine a lower bound to the travelling salesman problem for this graph.

Every night a street cleaner is tasked wiey 4b6ehb.dg* mi8* gkth cleaning the streets in his designated8* e dyb6kh m*gb.i4eg area.
The diagram below shows his designated area, with the weights indicating the time, in minutes, it takes to clean each street, and intersections indicated by letters.



The street cleaner must clean every street and wishes to minimise the time the task will take.
1. The street cleaner starts and finishes at the company depot, located at point K. Determine the total time it will take him to complete his task.

A new road is to be constructed directly from the intersection at point B to the intersection at point G, and it is estimated that this street will take 25 minutes to clean. The street cleaner has also been told he can now start the job at any intersection, and finish at any other intersection, and another employee will pick him up and drop him off.
2. Determine the difference in total time the job will take once the new road is completed.

The distances by road, in kilometres, between six citif4do ee gg:t t3oxo:/8es in the Netherlands are shown in the follow/gge4xo fd:t3 eoo: 8ting table.



A telecommunications company has gained approval to connect the six cities with fiber optic cabling along the road system.
1. Using Kruskal's algorithm, calculate the minimum length of fiber optic cable needed for all six cities to be connected to the fibre optic network.

Visitors to the Netherlands can travel between cities on a train network represented by the following graph.



2. State, justifying your answer, whether the graph has any Hamiltonian:
(1) paths;
(2) cycles.
3. State, justifying your answer, whether the graph has any Eulerian:
(1) trails;
(2) circuits.

Clare plans to visit the Netherlands. She wants to arrive in Eindhoven, travel all the available train routes exactly once, and then depart from Amsterdam.
4. Find the route between two cities Clare will need to travel by an alternate method of transport in order to make this possible.

The weights of the edges in th,l2-5p)qqaxf 2gw s yge complete graph $boldsymbol{G}$are given in the following table.


1. Starting at vertex A, use the nearest-neighbour algorithm to find an upper bound for the travelling salesman problem for graph $\boldsymbol{G}$.
2. By first deleting vertex A, use the deleted vertex algorithm together with Kruskal's algorithm to find a lower bound for the travelling salesman problem for $boldsymbol{G}$ .

  The table below shows the costs, in euros, of the direct buqq mm05rd 1-tgh; vwn0s rides between six towns (A-F). Cells with dashes indicate that there are no direct bus rt-gmq;n05 q0whmrd 1vides between the two towns.




1. Draw a weighted graph showing the direct bus rides between the towns and their costs.
2. (1) Write down the adjacency matrix for the graph of direct bus rides between the towns.
(2) Hence find the number of different ways to travel from and return to town F in exactly 6 bus rides.
3. State whether it is possible to travel from and return to town \mathrm{F} in exactly 6 bus rides, having visited each of the other 5 towns exactly once.
The following table shows the minimum cost to travel between the six towns. A travelling salesman wants to visit each of the six towns, starting and finishing at town F.



4. Find the value of p and the value of q .p =    q =   
5. Use the nearest-neighbour algorithm to find an upper bound for the cost of the travelling salesman's trip.    euros
6. By deleting vertex F , use the method of vertex deletion to find a lower bound for the cost of the travelling salesman's trip.   euros