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Questions and Answers
What is the main purpose of Prim's Algorithm?
What is the main purpose of Prim's Algorithm?
In Prim's Algorithm, how do you choose the next node to include in the tree?
In Prim's Algorithm, how do you choose the next node to include in the tree?
What is the main difference between Dijkstra's Algorithm and Prim's Algorithm?
What is the main difference between Dijkstra's Algorithm and Prim's Algorithm?
In Dijkstra's Algorithm, how do you calculate the shortest distance to the start for all nodes adjacent to the chosen node?
In Dijkstra's Algorithm, how do you calculate the shortest distance to the start for all nodes adjacent to the chosen node?
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What is the purpose of putting a line through the row of the chosen node in Prim's Algorithm for matrices?
What is the purpose of putting a line through the row of the chosen node in Prim's Algorithm for matrices?
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What is the stopping criterion for Dijkstra's Algorithm?
What is the stopping criterion for Dijkstra's Algorithm?
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What defines a 'simple' graph?
What defines a 'simple' graph?
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Which statement about 'walks' in a graph is true?
Which statement about 'walks' in a graph is true?
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What does the Handshaking Lemma state about a graph?
What does the Handshaking Lemma state about a graph?
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In the context of matrices, what do adjacency matrices represent?
In the context of matrices, what do adjacency matrices represent?
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Which property is true for the minimum spanning tree (MST) of a graph?
Which property is true for the minimum spanning tree (MST) of a graph?
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What occurs when two matrices A and B are multiplied?
What occurs when two matrices A and B are multiplied?
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How many edges does the minimum spanning tree (MST) for n nodes have?
How many edges does the minimum spanning tree (MST) for n nodes have?
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What does it mean if a walk is described as 'closed'?
What does it mean if a walk is described as 'closed'?
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Study Notes
Graphs
- A graph is a collection of points (nodes or vertices) connected by lines (edges).
- A simple graph has at most one edge between two nodes and no edges that loop back to the same vertex.
Walks, Paths, and Cycles
- A walk is a succession of connected edges of length n.
- 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 connected by a line.
- 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 degrees in a graph is twice the total number of edges.
Matrices
- Graph information can be stored in a matrix, which is a grid of numbers.
- Adjacency matrices store information about graph connections.
- Raising an adjacency matrix to a power n gives the number of walks of length n between nodes.
- Undirected graphs are symmetrical along their main diagonal.
- Distance matrices record the weight of each edge between nodes.
Matrix Multiplication
- Matrix multiplication is not commutative (AB ≠ BA).
- The product of two matrices at position m,n is the dot product of the mth row in A and the nth column in B.
The Minimum Spanning Tree
- A tree is a connected graph with no cycles that visits every node once.
- The minimum spanning tree (MST) is the tree-shaped subgraph with the lowest total weight.
- The MST for n nodes has n - 1 edges.
- 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
- 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.
Prim's Algorithm for Matrices
- Choose any 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 continue working with it.
- Repeat until the desired end node has a shortest distance written.
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Description
Learn about the definitions and concepts of graphs, walks, paths, and cycles in graph theory. Understand the difference between a walk, path, and cycle.
