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Questions and Answers
What characterizes a simple graph?
What characterizes a simple graph?
How does the Handshaking Lemma relate the sum of vertex degrees to edges in a graph?
How does the Handshaking Lemma relate the sum of vertex degrees to edges in a graph?
What defines a Minimum Spanning Tree (MST) in graph theory?
What defines a Minimum Spanning Tree (MST) in graph theory?
Which statement about paths and cycles is true?
Which statement about paths and cycles is true?
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What happens when two matrices are multiplied?
What happens when two matrices are multiplied?
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Consider a graph with nodes A, B, C, D, and E. Edge weights are as follows: AB=2, AC=4, AD=3, AE=5, BC=1, BD=6, BE=7, CD=2, CE=3, DE=4. Using Prim's Algorithm, which edge would be selected after the edge AC?
Consider a graph with nodes A, B, C, D, and E. Edge weights are as follows: AB=2, AC=4, AD=3, AE=5, BC=1, BD=6, BE=7, CD=2, CE=3, DE=4. Using Prim's Algorithm, which edge would be selected after the edge AC?
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Suppose we are using Dijkstra's Algorithm to find the shortest path from node A to node E in a graph. We have already determined the shortest distances from A to B and A to C. Which node would we choose to expand from next?
Suppose we are using Dijkstra's Algorithm to find the shortest path from node A to node E in a graph. We have already determined the shortest distances from A to B and A to C. Which node would we choose to expand from next?
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Consider a graph with nodes A, B, C, D, and E, and edge weights as follows: AB=2, AC=4, AD=3, AE=5, BC=1, BD=6, BE=7, CD=2, CE=3, DE=4. Which of the following statements about Prim's Algorithm applied to this graph is FALSE?
Consider a graph with nodes A, B, C, D, and E, and edge weights as follows: AB=2, AC=4, AD=3, AE=5, BC=1, BD=6, BE=7, CD=2, CE=3, DE=4. Which of the following statements about Prim's Algorithm applied to this graph is FALSE?
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In Dijkstra's Algorithm, how do we handle negative edge weights in a graph?
In Dijkstra's Algorithm, how do we handle negative edge weights in a graph?
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Which of the following statements accurately describes the difference between Prim's Algorithm and Dijkstra's Algorithm?
Which of the following statements accurately describes the difference between Prim's Algorithm and 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.
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Description
Learn the fundamental concepts of graph theory, including vertices, edges, walks, paths, and cycles. Understand the differences between a walk and a path, and how to identify a cycle.
