Questions and Answers
What does the Handshaking Lemma state about the relationship between the degrees of vertices and the number of edges in a graph?
How is the degree of a vertex defined in a graph?
Which of the following is a characteristic of a simple graph?
What is the difference between a walk and a path in a graph?
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What information does an adjacency matrix of a graph store?
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What is the significance of raising an adjacency matrix to the power n?
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How is the minimum spanning tree (MST) of a graph defined?
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What is the main difference between Kruskal's Algorithm and Prim's Algorithm for finding the MST?
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What is the primary objective of Prim's Algorithm in graph theory?
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In Dijkstra’s Algorithm, what is the first action you take after selecting the starting node?
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Which step is NOT part of Prim's Algorithm for constructing a minimum spanning tree?
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What does Dijkstra’s Algorithm primarily compute?
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What process is repeated in both Prim's Algorithm and Dijkstra's Algorithm?
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When implementing Dijkstra’s Algorithm, which action must be taken after writing down the shortest distances?
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What is the main purpose of Prim's Algorithm?
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What is the criteria for choosing the next node in Prim's Algorithm?
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What is the primary difference between Dijkstra's Algorithm and Prim's Algorithm?
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What is the purpose of circling the edge of least weight in the column of the chosen node in Prim's Algorithm for matrices?
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What is the condition for stopping the iteration in Dijkstra's Algorithm?
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Study Notes
Graphs
- A graph is a collection of points (nodes or vertices) connected by lines (edges).
- A graph is simple if it has at most one edge between two nodes and no edges that loop back to the same vertex.
Walks and Paths
- A walk of length n is a sequence of n connected edges.
- A path is a walk that does not visit the same node twice.
- A cycle is a path that ends at the same node it started at.
- A walk is closed if it ends at the same node it started at, and open otherwise.
- Two nodes are adjacent if they are joined by an edge.
- A node and edge are incident if they touch.
The Handshaking Lemma
- The degree of a vertex is the number of edges that meet at that vertex.
- The Handshaking Lemma states that the sum of all vertex degrees in a graph is twice the total number of edges.
Matrices
- Graph information can be stored in a grid of numbers called a matrix.
- Adjacency matrices store information about graph connections.
- Raising an adjacency matrix to a power n gives the number of walks of length n between any two nodes.
- Undirected graphs have symmetrical adjacency matrices along their main diagonal.
Distance Matrices
- Distance matrices record the weight of each edge between nodes.
- Matrix multiplication is used to calculate shortest paths.
Matrix Multiplication
- When two matrices A and B multiply, the product matrix position m,n is the dot product of the mth row in A and the nth column in B.
- Matrix multiplication is not commutative: AB ≠ BA.
The Minimum Spanning Tree
- A tree is a connected graph with no cycles, covering every node once.
- The minimum spanning tree (MST) of a graph is the tree-shaped subgraph with the lowest total weight.
- The MST for n nodes has n - 1 edges.
Finding the Minimum Spanning Tree
- There are two algorithms to find the MST: Kruskal's Algorithm and Prim's Algorithm.
Kruskal's Algorithm
- Pick the edge of least weight in the graph.
- Select the next edge of least weight that does not create a cycle.
- Repeat until all nodes are included.
Prim's Algorithm
- For graphs:
- Choose any node.
- Choose the edge of least weight connected to this node.
- Choose the next lowest weight edge connecting the tree to a new node.
- Repeat until all nodes are included.
- For matrices:
- Choose a starting node.
- Put a line through the row of the chosen node.
- Highlight the column of the chosen node.
- Circle the edge of least weight in the column.
- Repeat until all rows are deleted.
Dijkstra's Algorithm
- Dijkstra's algorithm finds the shortest path between any two nodes.
- Steps:
- Begin at the starting node.
- Calculate the shortest distance back to the start for all adjacent nodes.
- Choose the node with the shortest distance and repeat the process.
- Repeat until the desired end node has a shortest distance written.
Graphs
- A graph is a collection of points (nodes or vertices) connected by lines (edges).
- A graph is simple if it has at most one edge between two nodes and no edges that loop back to the same vertex.
Walks and Paths
- A walk of length n is a sequence of n connected edges.
- A path is a walk that does not visit the same node twice.
- A cycle is a path that ends at the same node it started at.
- A walk is closed if it ends at the same node it started at, and open otherwise.
- Two nodes are adjacent if they are joined by an edge.
- A node and edge are incident if they touch.
The Handshaking Lemma
- The degree of a vertex is the number of edges that meet at that vertex.
- The Handshaking Lemma states that the sum of all vertex degrees in a graph is twice the total number of edges.
Matrices
- Graph information can be stored in a grid of numbers called a matrix.
- Adjacency matrices store information about graph connections.
- Raising an adjacency matrix to a power n gives the number of walks of length n between any two nodes.
- Undirected graphs have symmetrical adjacency matrices along their main diagonal.
Distance Matrices
- Distance matrices record the weight of each edge between nodes.
- Matrix multiplication is used to calculate shortest paths.
Matrix Multiplication
- When two matrices A and B multiply, the product matrix position m,n is the dot product of the mth row in A and the nth column in B.
- Matrix multiplication is not commutative: AB ≠ BA.
The Minimum Spanning Tree
- A tree is a connected graph with no cycles, covering every node once.
- The minimum spanning tree (MST) of a graph is the tree-shaped subgraph with the lowest total weight.
- The MST for n nodes has n - 1 edges.
Finding the Minimum Spanning Tree
- There are two algorithms to find the MST: Kruskal's Algorithm and Prim's Algorithm.
Kruskal's Algorithm
- Pick the edge of least weight in the graph.
- Select the next edge of least weight that does not create a cycle.
- Repeat until all nodes are included.
Prim's Algorithm
- For graphs:
- Choose any node.
- Choose the edge of least weight connected to this node.
- Choose the next lowest weight edge connecting the tree to a new node.
- Repeat until all nodes are included.
- For matrices:
- Choose a starting node.
- Put a line through the row of the chosen node.
- Highlight the column of the chosen node.
- Circle the edge of least weight in the column.
- Repeat until all rows are deleted.
Dijkstra's Algorithm
- Dijkstra's algorithm finds the shortest path between any two nodes.
- Steps:
- Begin at the starting node.
- Calculate the shortest distance back to the start for all adjacent nodes.
- Choose the node with the shortest distance and repeat the process.
- Repeat until the desired end node has a shortest distance written.
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Description
Learn about the fundamentals of graphs, including nodes, edges, walks, paths, and cycles. Understand the differences between each concept and how they relate to each other.
