Graph Theory Basics

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Questions and Answers

What is the main purpose of Prim's Algorithm?

To find the longest path in a graph

To find the maximum spanning tree of a graph

To find the shortest path between two nodes

To find the minimum spanning tree of a graph (correct)

In Prim's Algorithm, how do you choose the next node to include in the tree?

Choose the node with the lowest degree

Choose the node connected to the current node by the edge of least weight (correct)

Choose the node with the highest degree

Choose the node furthest from the current node

What is the main difference between Dijkstra's Algorithm and Prim's Algorithm?

Dijkstra's Algorithm is used for finding the shortest path, while Prim's Algorithm is used for finding the minimum spanning tree (correct)

Dijkstra's Algorithm is used for weighted graphs, while Prim's Algorithm is used for unweighted graphs

Dijkstra's Algorithm is used for finding the longest path, while Prim's Algorithm is used for finding the maximum spanning tree

Dijkstra's Algorithm is used for directed graphs, while Prim's Algorithm is used for undirected graphs

In Dijkstra's Algorithm, how do you calculate the shortest distance to the start for all nodes adjacent to the chosen node?

By adding the weight of the edge to the current node Signup and view all the answers

What is the purpose of putting a line through the row of the chosen node in Prim's Algorithm for matrices?

To delete the row of the chosen node Signup and view all the answers

What is the stopping criterion for Dijkstra's Algorithm?

When the desired end node has a shortest distance written Signup and view all the answers

What defines a 'simple' graph?

A graph with at most one edge connecting two nodes and no loops. Signup and view all the answers

Which statement about 'walks' in a graph is true?

A walk allows visiting the same vertex multiple times. Signup and view all the answers

What does the Handshaking Lemma state about a graph?

The sum of all degrees in a graph is twice the total number of edges. Signup and view all the answers

In the context of matrices, what do adjacency matrices represent?

The direct connections between nodes in the graph. Signup and view all the answers

Which property is true for the minimum spanning tree (MST) of a graph?

It is a connected graph with no cycles, hitting every node once. Signup and view all the answers

What occurs when two matrices A and B are multiplied?

Position m,n in the product matrix represents their dot product. Signup and view all the answers

How many edges does the minimum spanning tree (MST) for n nodes have?

n - 1 edges Signup and view all the answers

What does it mean if a walk is described as 'closed'?

It ends at the same node where it began. Signup and view all the answers

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.