Graph Theory Basics Quiz

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Questions and Answers

What distinguishes a graph in graph theory from a graph of a function?

Graph theory graphs are used for discrete representations, where function graphs deal with continuous data.

Graph theory graphs represent numerical values, whereas function graphs show relationships between variables.

Graph theory graphs consist of vertices and edges, while function graphs depict curves on a coordinate plane. (correct)

Graph theory graphs model relationships between objects, but function graphs represent mathematical formulas.

In graph theory, what defines an edge?

A directed path connecting a source to a destination.

An unordered pair of vertices. (correct)

A single point connecting two other points.

A path between two points with a sequence of intermediate points.

If a graph has an edge that connects a vertex to itself, what type of edge is it?

Isolated edge

Pendant edge

Loop (correct)

Parallel edge

What term describes multiple edges that connect the same pair of vertices?

Parallel edges Signup and view all the answers

Using the Handshaking Lemma, if a graph has 5 vertices, each with a degree of 4, how many edges are in the graph?

10 Signup and view all the answers

What is a vertex with a degree of zero called?

Isolated vertex Signup and view all the answers

What is a graph called if it has no loops and no parallel edges?

Simple graph Signup and view all the answers

What does the theorem of even number of odd vertices imply in any given graph?

If there are any odd degree vertices, their number must be even. Signup and view all the answers

In a graph, if the sum of degrees of vertices with an odd degree is $X$, and the sum of degrees of vertices with an even degree is $Y$, and the total sum of all degrees is $Z$, which of the following is always true?

$X$ is even, $Y$ is even, and $Z$ is even. Signup and view all the answers

A network of telephone lines and poles is represented by a graph. What does it mean if removing a single edge disconnects the graph?

The edge represents a critical line. Signup and view all the answers

Which type of graph has no edges?

Null Graph Signup and view all the answers

In a complete graph with $n$ vertices, how many edges are there?

$\frac{n(n-1)}{2}$ Signup and view all the answers

A graph where every vertex has the same degree $k$ is called a:

Regular graph Signup and view all the answers

If a graph's vertex set can be divided into two subsets such that every edge connects vertices from different subsets, what type of graph is it?

Bipartite Graph Signup and view all the answers

If H is a subgraph of G and H contains all vertices of G, then H is called a:

Spanning Subgraph Signup and view all the answers

If a graph $G$ has edges and an edge $e$ is removed from them, what kind of subgraph is formed?

Edge Deleted Subgraph Signup and view all the answers

What characterizes a trail in a graph?

It consists of a sequence of edges and vertices with no repeated edges. Signup and view all the answers

Which statement is true regarding a connected graph?

All vertices are part of only one connected component. Signup and view all the answers

What does the vertex induced subgraph G consist of?

Only the vertices in subset U and edges between them. Signup and view all the answers

In graph theory, what is a cycle?

A path that returns to the starting vertex without repeating any others. Signup and view all the answers

How is an edge deleted subgraph G - e4 defined?

A subgraph including all vertices but removing one edge. Signup and view all the answers

Which of the following defines a disconnected graph?

A graph where there is at least one pair of vertices that are not connected. Signup and view all the answers

What is the result of the theorem regarding the complement of a simple graph G?

If G is disconnected, then its complement G is connected. Signup and view all the answers

What is represented by C(u) in a graph?

The collection of all vertices connected to vertex u. Signup and view all the answers

What condition must be true for a graph to be considered connected based on the partition of its vertex set?

There exists at least one edge connecting two vertices from different subsets. Signup and view all the answers

According to the proof of Theorem 4, what happens when m equals zero?

The number of components equals the number of vertices. Signup and view all the answers

What does the proof of Theorem 5 indicate about a graph with exactly two vertices of odd degree?

There exists a path connecting these two vertices. Signup and view all the answers

In the context of disconnected graphs, what conclusion can be drawn if there are no edges connecting vertex sets G₁ and G₂?

The entirety of G is a disjoint union. Signup and view all the answers

What is the relationship between the number of vertices n, components k, and edges m in a simple graph as presented in Theorem 4?

n - k ≤ m. Signup and view all the answers

What is a critical aspect of the proof technique used in Theorem 4?

It uses induction based on the number of edges. Signup and view all the answers

What implication does a graph with k components have on the existence of edges in the graph?

At least n - k edges must exist. Signup and view all the answers

What defines the complement of a graph G?

It includes edges that are present in G but not in G. Signup and view all the answers

Which of the following statements about the components of a graph is true?

For any two vertices in separate components, there can be no path connecting them. Signup and view all the answers

What was concluded by Euler regarding the Konigsberg Seven Bridge Problem?

It is impossible to do so based on the even-odd degree rule. Signup and view all the answers

In a bipartite graph, which statement is accurate regarding the lengths of cycles?

All cycles in the graph must have even length. Signup and view all the answers

What condition must hold true for a graph to be self-complementary?

The number of vertices must be of the form 4k or 4k + 1. Signup and view all the answers

What defines a connected graph?

All pairs of vertices must have at least one path connecting them. Signup and view all the answers

In the context of graph theory, what does W(G) represent?

The number of connected components in graph G. Signup and view all the answers

Which method can show that a graph with even-length cycles is bipartite?

Defining distances from an arbitrary vertex. Signup and view all the answers

What characteristic is unique to a self-complementary graph's vertices?

The number of edges is equal to that of its complement. Signup and view all the answers

Study Notes

Graph Theory Introduction

Graph theory involves studying vertices and edges
Vertices represent objects and edges represent connections between those objects
Applications include: acquaintance, chemical bonding, electrical networks, transportation networks, and binary vectors
Techniques used include induction, parity, extremality, counting, and inclusion-exclusion methods

Graph Definition

A graph G = (V, E) consists of a set of vertices (V) and a set of edges (E)
Vertices are the points, and edges connect these points
Each edge is associated with an unordered pair of vertices (end vertices)

Elementary Terminologies

End vertices of an edge: The vertices an edge connects
Adjacent vertices: Vertices connected by an edge
Loop: An edge with both end vertices identical
Parallel (multiple) edges: Two or more edges connecting the same pair of vertices
Degree (valency) of a vertex: The number of edges incident to that vertex (Loops contribute 2 to the degree)
Isolated vertex: A vertex with degree 0
Pendent vertex: A vertex with degree 1
Simple graph: A graph with no loops or parallel edges
Multigraph: A graph with loops or parallel edges

Handshaking Lemma

In any graph, the sum of the degrees of all vertices is equal to twice the number of edges
Σd(v) = 2m (where m is the number of edges)

Odd and Even Vertices

Odd vertex: A vertex with an odd degree
Even vertex: A vertex with an even degree
In any graph, the number of odd vertices is even

Special Types of Graphs

Null graph (Nₙ): A graph with no edges
Complete graph (Kₙ): A graph where every pair of vertices is connected by an edge
Note that K₁: 1 vertex, K₂: 2 vertices, K3.3: 2 sets of vertices with every vertex in one set connected to every vertex in the other set

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Description

Test your knowledge of fundamental concepts in graph theory. This quiz covers critical definitions and properties, such as edges, vertices, loops, and the Handshaking Lemma. Challenge yourself to understand the distinctions between different types of graphs and their characteristics.