Introduction to Graph Theory

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Questions and Answers

What is the correct definition of a graph in graph theory?

A collection of nodes and edges. (correct)

A collection of edges and weights only.

A collection of edges and directions.

A collection of nodes and vertices only.

Which of the following correctly describes a neighborhood in graph theory?

The total number of nodes in the graph.

The edge set connected to a node.

The set of nodes connected to a specific node via edges. (correct)

The collection of all directed edges in a graph.

Which statement correctly describes a directed graph?

Edges can connect any two vertices without direction.

All edges have no direction.

Vertices do not have connections between them.

Edges have a source and a target vertex. (correct)

What distinguishes a weighted graph from other types of graphs?

Each edge is associated with a numeric value (weight). Signup and view all the answers

What is the correct notation to denote the set of vertices in a graph G?

V(G) Signup and view all the answers

In the context of graph theory, what does a null graph represent?

A graph with no edges and no vertices. Signup and view all the answers

Which characteristic accurately describes an undirected graph?

Edges have no specific direction. Signup and view all the answers

Which of the following best describes a loop in graph theory?

An edge connecting a vertex to itself. Signup and view all the answers

What defines a tree in an undirected graph?

It does not contain a cycle and is connected. Signup and view all the answers

What characterizes adjacent nodes in a graph?

They are connected via an edge. Signup and view all the answers

In the context of directed graphs, how is the degree of a vertex determined?

By adding the number of incoming and outgoing edges. Signup and view all the answers

What does an isolated vertex mean in graph theory?

It is not connected to any other vertex. Signup and view all the answers

What distinguishes a 'walk' from a 'trail' in graph terminology?

A trail allows repeated edges but not nodes. Signup and view all the answers

How is a closed path defined in graph theory?

It is a path where the start and end vertices are the same. Signup and view all the answers

If a vertex has a degree of 0, what can be said about it?

It is an isolated vertex. Signup and view all the answers

What outcome is true if there are two vertices with more than one path connecting them in a tree?

This results in a cycle within the tree. Signup and view all the answers

Study Notes

History of Graph Theory

Graphs are used to model pairwise relationships between objects
The study of graphs has been around for centuries, with its origins traced back to Euler's work on the Königsberg bridge problem in the 18th century.

Basic Concepts of Graph Theory

A graph is a collection of nodes (vertices) and edges that represent connections between those nodes.
Graphs are often represented using the notation G = (V, E), where V is the set of vertices and E is the set of edges.

Graph Representations

Graphs can be represented in various ways, including adjacency matrices, adjacency lists, and incidence matrices.

Graph Terminologies

Vertex/Node: Basic element of a graph, represented as a dot or node.
Edge/Arc: A connection between two vertices, represented as a line.
Neighborhood: The set of nodes connected to a given node via an edge.

Different Types of Graphs

Undirected Graph: Edges have no direction.
Directed Graph (Digraph): Edges have direction, represented by arrows.
Weighted Graph: Edges have associated numerical values (weights).
Empty Graph: No edges between vertices.
Null Graph: No nodes or edges.
Tree: Connected and acyclic undirected graph, with a unique path between any two nodes.

Undirected Graph

Edges represent relationships without direction
Edges are unordered pairs of nodes
Example: Friendship network, where edges represent friendships between people
Example: A graph representing friendships, with people as vertices and edges connecting friends.

Directed Graph

Edges have direction
Edges are ordered pairs of nodes
Example: Edges represent one-way relationships, such as followers on social media or roads where travel is only possible in one direction.
Example: A graph depicting a social network where edges represent "follows" relationships, indicating who follows whom.

Weighted Graph

Edges have numerical values representing weights
Weights represent factors like distances, costs, or capacities
Example: A graph representing cities, where edges represent distances between them with weights representing kilometers.

Connected and Isolated Vertex

Connected: Two vertices are connected if there is a path between them.
Isolated Vertex: A vertex that is not connected to any other vertex.
Example: In a social network graph, an isolated vertex would represent an individual without any connections.

Adjacent Nodes

Two nodes are adjacent if they are connected by an edge.
An edge where the two end vertices are the same is called a loop or a self-loop.
Example: In a friendship graph, two people are adjacent if they are friends.

Degree of a Vertex

Degree (Undirected Graphs): The number of edges incident on a node.
Degree (Directed Graphs):
- In-degree: The number of edges entering the node.
- Out-degree: The number of edges leaving the node.
Total degree: The sum of in-degree and out-degree.
Example: In a directed graph representing a network of roads, the in-degree of a vertex representing a city represents the number of roads leading into the city, while the out-degree represents the number of roads leading out of the city.

Walk, Trail, & Path

Walk: A sequence of nodes and edges, where edges and nodes may be repeated.
Trail: A walk where no edge is repeated, but nodes may be repeated.
Path: A trail where no vertex is repeated.
Cycle: A closed path.
Example: In a map representing cities and roads, a walk could be a route that visits some cities and roads multiple times, a trail could be a route that visits each road only once but may return to a city, and a path would be a route that visits each city and road only once.

Classification of Graph Terms

Global Terms: Refer to the whole graph.
Local Terms: Refer to a specific node in a graph.
Example: Graph diameter is a global term that refers to the longest shortest path between any two vertices in a graph. The degree of a vertex is a local term that describes the number of edges connected to a specific node.

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Description

Explore the fundamentals of graph theory, including its history, basic concepts, and various representations. This quiz covers essential terminologies and different types of graphs, enhancing your understanding of how graphs model relationships between objects.