Graph Theory and Project Scheduling Concepts

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Questions and Answers

Which activity has no immediate predecessors?

A (correct)

What is the immediate predecessor of activity I?

E (correct)

Which activity does NOT lead directly to the finish?

G (correct)

What is a characteristic of a directed graph in this scenario?

Edges represent one-way paths. Signup and view all the answers

Which statement about the inverse of matrices is true?

If the determinant of a matrix is equal to zero, then the inverse does not exist. Signup and view all the answers

In the Leslie matrix for the bird species, what does the element in the second row, first column represent?

The average proportion of females that survive into their second year. Signup and view all the answers

What is indicated by the zero values in the Leslie matrix regarding reproduction?

Females do not produce offspring in their first year. Signup and view all the answers

How many of the following statements about square matrices are true: All square matrices have an inverse; the inverse could be the same as the transpose; the determinant must be non-zero for an inverse to exist; an identity matrix has an inverse?

2 Signup and view all the answers

Considering the life span and reproduction of the bird species, how many offspring do females produce on average over their lifespan?

7 Signup and view all the answers

What is the weight of the minimum spanning tree given that the edge weights are 15, 10, 8, 12, 9, and 7?

32 Signup and view all the answers

What is the minimum total time taken to complete the duties for Anthea, Bob, Cho, and Dario?

31 Signup and view all the answers

How many edges does the planar graph represented by the adjacency matrix have?

9 Signup and view all the answers

Based on the adjacency matrix, how many vertices are connected to vertex K?

3 Signup and view all the answers

If a new activity is added that connects to activities with the highest total time usage, what will be the impact on total project completion time?

It will increase the total time Signup and view all the answers

What is the maximum time taken by any one employee to complete their assigned duty?

10 Signup and view all the answers

In the planar graph with 5 vertices, if the number of edges is 10, how many faces are there according to Euler's formula?

7 Signup and view all the answers

How do the completion times of Cho and Anthea for Duty 3 compare?

Cho is faster than Anthea Signup and view all the answers

What does element m21 in the matrix represent?

The temperature was 29 °C on Monday in week 2. Signup and view all the answers

Which temperature is reported for Wednesday in week 1?

23 °C Signup and view all the answers

How many days did the temperature reach 20 °C or higher in week 2?

5 days Signup and view all the answers

Which three letters change position when Q is multiplied by P?

e, a, m Signup and view all the answers

Which location will have no birds remaining in the long term?

Location M Signup and view all the answers

What is the average maximum temperature for week 1?

19.14 °C Signup and view all the answers

Which day had the highest temperature recorded

Sunday of week 1 Signup and view all the answers

If the temperatures for week 2 are considered, what was the lowest recorded temperature?

20 °C Signup and view all the answers

What is the maximum flow capacity from entrance A to exit H using the given paths?

24 children Signup and view all the answers

If the capacity from edge CD is increased by 5, what will be the new capacity of flow from entrance to exit?

28 children Signup and view all the answers

Which edge currently has the smallest capacity limiting the flow from entrance to exit?

Edge FG Signup and view all the answers

What would be the effect of reversing the direction of flow along edge FG?

It would decrease the overall capacity. Signup and view all the answers

If capacity along edge BH is increased to 15, what would be the total flow capacity from A to G?

30 children Signup and view all the answers

How many total paths can children take from A to exit H based on current edge capacities?

4 distinct paths Signup and view all the answers

Which edge alteration would provide the most significant capacity increase?

Increasing capacity on GH to 20 Signup and view all the answers

What is the significance of a cut in this network regarding flow management?

It identifies maximum flow limits. Signup and view all the answers

Study Notes

Minimum Spanning Tree

The graph has 7 vertices (points) and 8 edges (lines) connected between them
The minimum spanning tree is the tree with the lowest total weight (sum of all the numbers)
The minimum spanning tree is found by connecting all the vertices with the smallest possible weights
The weight of the minimum spanning tree is 30

Employee Duty Allocation

Anthea, Bob, Cho and Dario are assigned duties 1, 2, 3 and 4
The manager tries to allocate duties to minimise the total time to complete all duties
The minimum total time is 29 minutes

Graph Planarity

The adjacency matrix represents a graph
The matrix displays connection between vertices
The number of faces in the graph is 7

Project Scheduling

The project has 10 activities to be completed
The project network is a directed graph
The directed graph shows the dependencies of activities and their predecessors
The graph represents a sequence of activities with arrows connecting predecessors to successors
Option D correctly shows the project's activity dependencies using a directed graph

Flow Network

The network shows one-way paths through a maze
Paths are from the entrance (A) to exit (H)
Network has vertices (intersections) and connecting edges (paths)
The capacity of a cut is the sum of capacities of edges across the cut
The minimum cut capacity is the smallest possible capacity of all cuts
The minimum cut capacity is 23
Increasing the capacity of edge GH to 16 would lead to the greatest increase in flow

Matrix Operations

Element m21 is the temperature for Week 2, Monday
Matrix P is a permutation matrix, and matrix Q is a column matrix
Multiplying Q by P rearranges the letters in Q
Letters e, a and s change position

Markov Chains

The transition matrix T models movement of birds between locations
The rows of the transition matrix represent the starting location
The columns represent the ending location
The long-term outcome is that the number of birds will decrease at locations M and N and increase at location O

Matrix Inverses

There are 4 statements about true and false statements regarding matrix inverses
Two of the statements are true
Not all square matrices have inverses
The inverse of a matrix could be the same as the transpose of that matrix, but not always

Leslie Matrices

The Leslie matrix models population distribution of a species
Each column of the matrix represents an age class
Each row represents the rate of change for that age class
The element in the second row, first column represents the survival rate from the first year to the second year

School Lunch Transitions

The transition diagram shows the proportion of students moving between four locations
Numbers on the edges represent the maximum number of students allowed to move along that edge per minute
The entrance and exit nodes allow for students to enter and leave the system
The network could be used to track the flow of students through the different locations

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Description

This quiz covers essential topics in graph theory, including minimum spanning trees and graph planarity, as well as practical applications in project scheduling. Explore concepts such as duty allocation and project dependencies to deepen your understanding of efficient graph utilization.