Math 20223 Graph Theory Overview

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Questions and Answers

What does it mean when two vertices are said to be adjacent?

They share an edge in the graph. (correct)

They are at the opposite ends of the graph.

They are part of different graphs.

They have no connections to each other.

If vertex u is incident to edge a, which statement is true?

Vertex u connects only to vertex v.

Vertex u cannot connect to any other edges.

Vertex u must be one of the endpoints of edge a. (correct)

Vertex u is not relevant to the graph structure.

Which of the following lists are all the ends of one edge in a graph?

u, x, y and b

u, v, x, and y

x and y

u and v (correct)

How would you describe the relationship between incident vertices and edges?

All vertices of a graph are incident to at least one edge. Signup and view all the answers

What would be a valid exercise related to identifying edges in a graph?

Identify all adjacent vertices to a specific vertex. Signup and view all the answers

What is the neighborhood of the vertex v2 in graph G?

{v4, v5} Signup and view all the answers

Which statement correctly describes the degree of vertex v3?

deg(v3) = 3 Signup and view all the answers

What is the neighborhood of vertex x in graph H?

{u, v, w, y} Signup and view all the answers

How many vertices are in the neighborhood of vertex u?

3 Signup and view all the answers

If deg(v2) = 2, which vertices are directly connected to v2?

v5 and v4 Signup and view all the answers

What does ν(G) represent in a graph?

The number of vertices in graph G Signup and view all the answers

If ε(G) = 8, what does this signify about the graph G?

G has 8 edges Signup and view all the answers

What can be inferred about a vertex that is part of a loop in a graph?

It forms a cycle with itself Signup and view all the answers

In the context of graph theory, what is a neighborhood of a vertex?

All vertices connected by edges to that vertex Signup and view all the answers

Given that ν(G) = 5 and ε(G) = 8, what can be deduced about the graph's connectivity?

The graph may be disconnected Signup and view all the answers

What is one of the key elements of a graph?

Edges that connect vertices Signup and view all the answers

In graph theory, what does joining a vertex to itself represent?

A loop or self-edge Signup and view all the answers

Which of the following statements about edges in a graph is NOT true?

Edges can only connect two different vertices Signup and view all the answers

What fundamental problem related to graph theory did Leonard Euler address?

The Konigsberg Bridges Problem Signup and view all the answers

What type of edge connects two different vertices in a graph?

Undirected edge Signup and view all the answers

Study Notes

Course Information

Course title: Math 20223 Graph Theory with Application
Instructor: Rafael A. Duarte
Department: Mathematics and Statistics
Date: Thursday, November 14th, 2024

Graph Theory Overview

Graphs are mathematical structures consisting of vertices and edges
A graph is an ordered triple (V(G), E(G), ΨG)
V(G) is a set of vertices
E(G) is a set of edges
ΨG is an incidence function
Vertices represent objects, and edges represent relationships between them
The Königsberg Problem, solved by Leonard Euler, marked the start of graph theory in the 18th Century.

Definitions

Vertex: A point or node in a graph. Also known as a node.
Edge: A line or connection between two vertices. Also known as an arc or line.
Loop: An edge that connects a vertex to itself.
Link: An edge that connects two distinct vertices.
Incident: A vertex is incident to an edge if the edge connects to the vertex
Adjacent: Two vertices are adjacent if there is an edge connecting them.
Order of a graph: The number of vertices
Size of a graph: The number of edges
Neighborhood of a vertex: The set of all vertices adjacent to that vertex
Degree of a vertex: The number of edges connected to that vertex
Isolated vertex: A vertex with a degree of 0
Dominating/universal vertex: A vertex with a degree n-1, where n is the order of the graph

Example Graphs (Refer to diagrams)

Examples demonstrate various graphs and illustrate different graph features
Specific examples provided and analyzed for graph elements (edges, vertices, loops, etc.)
Detailed analysis of the relationships between vertices and edges in a graph is shown within the notes

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Description

This quiz explores the foundations of Graph Theory, focusing on the essential concepts such as vertices, edges, loops, and the historical context including the Königsberg Problem. Designed for students of Math 20223, it will assess your knowledge of these critical mathematical structures and their applications.