Graphs Course Quiz

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is an Euler path in graph theory?

A path that uses each edge precisely once. (correct)

A path that visits each vertex exactly once.

A cycle that returns to the starting point using each vertex once.

A path that crosses every bridge in a graph at least once.

Which condition indicates that a graph has an Euler circuit?

All vertices have an even degree. (correct)

At least one vertex has an even degree.

All edges are connected in a single path.

All vertices have an odd degree.

What conclusion did Leonard Euler reach about the Seven Bridges of Königsberg?

Starting and finishing at different nodes allows a successful traversal.

It is impossible to traverse each edge precisely once. (correct)

The graph is Eulerian and has multiple Euler circuits.

A path exists that traverses each edge exactly once.

What defines a graph as traversable?

It contains an Euler circuit. Signup and view all the answers

Which statement is true regarding Euler paths and circuits?

Every Euler circuit must contain an Euler path. Signup and view all the answers

What defines a mixed graph?

A graph with both directed and undirected edges. Signup and view all the answers

How do you determine if two vertices are adjacent?

If they are connected by an edge. Signup and view all the answers

What is the definition of the degree of a vertex?

The total number of edges incident on a vertex. Signup and view all the answers

What does in-degree indicate for a vertex?

The number of edges for which the vertex is the terminal vertex. Signup and view all the answers

What does the neighborhood N(v) of a vertex v comprise?

All vertices that are adjacent to v. Signup and view all the answers

If a vertex has a degree of 3, how many edges are incident on it?

3 edges. Signup and view all the answers

If deg+(u) = 2 and deg-(u) = 0, what can be inferred about vertex u?

It has no outgoing edges. Signup and view all the answers

For the edge (u, v), which description is accurate?

Both u and v are endpoints of the edge. Signup and view all the answers

What defines the edges in a directed graph?

Edges are represented as an ordered pair of vertices. Signup and view all the answers

Which of the following best describes a self-loop in graph terminology?

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

In graph terminology, what are multiple edges?

Two or more edges joining the same pair of vertices. Signup and view all the answers

Which notation represents an undirected edge between two vertices u and v?

{u, v} Signup and view all the answers

What characterizes an undirected graph compared to a directed graph?

Edges treat both endpoints as interchangeable. Signup and view all the answers

When defining a graph G = (V, E), what do V and E specifically represent?

V is a set of vertices; E is a set of edges. Signup and view all the answers

Which of the following represents a directed edge from vertex u to vertex v?

(u, v) Signup and view all the answers

Which statement accurately applies to a graph composed of vertices and edges?

Graphs can include self-loops that connect a vertex to itself. Signup and view all the answers

What is the degree of vertex 'b' in the graph described?

6 Signup and view all the answers

Which of the following vertices is considered pendant?

c Signup and view all the answers

What is an isolated vertex?

A vertex with no edges Signup and view all the answers

According to the Handshaking Theorem, how many edges does a graph with 10 vertices each of degree 6 have?

30 Signup and view all the answers

Is it possible for a graph with 5 vertices to have each vertex with a degree of 3?

No, the total degree sum would be odd Signup and view all the answers

What is the degree of vertex 'k' in the provided graph structure?

0 Signup and view all the answers

In the context of the Handshaking Theorem, what does 'm' represent?

The number of edges Signup and view all the answers

If vertex 'd' has a degree of 5, how many vertices is it directly connected to?

5 Signup and view all the answers

What is a graph invariant?

A property that is preserved under isomorphism. Signup and view all the answers

Which of the following is a graph invariant?

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

If two graphs have different numbers of edges, what can be concluded?

The graphs are definitely not isomorphic. Signup and view all the answers

When determining if two graphs are isomorphic, which property cannot be ignored?

The maximal degree of vertices. Signup and view all the answers

Why were the graphs G and H not isomorphic in the example provided?

Their subgraphs of degree three were not isomorphic. Signup and view all the answers

What does the function f represent in the context of graph isomorphism?

A mapping of vertices from one graph to another. Signup and view all the answers

What characteristic explains why determining graph isomorphism can be challenging?

The best algorithms have exponential time complexity. Signup and view all the answers

What is the minimal degree denoted as d(G) in graph theory?

The smallest number of edges connected to any vertex. Signup and view all the answers

What is required for two graphs to be considered isomorphic?

They have a one-to-one and onto function between their vertex sets. Signup and view all the answers

Which of the following best describes an isomorphism?

A bijective function with adjacency preservation. Signup and view all the answers

Why is it difficult to determine if two graphs are isomorphic using brute force?

The number of possible correspondences grows factorially with the number of vertices. Signup and view all the answers

In the context of graph isomorphism, what is the significance of having the same vertex and edge counts?

It is a necessary condition for determining isomorphism but not sufficient. Signup and view all the answers

Which application utilizes graph isomorphism in its process?

Modeling chemical compounds. Signup and view all the answers

What can be inferred if two graphs have the same degree count for all vertices?

They might be isomorphic, but additional checks are needed. Signup and view all the answers

Why might electronic circuits be modeled as graphs?

To visually represent circuits and check connectivity. Signup and view all the answers

If two graphs are found to be not isomorphic, what can be concluded about their vertex degree distributions?

There must be at least one degree that does not match between the two graphs. Signup and view all the answers

Study Notes

Course Information

Course: Graphs
Instructor: Dr. Lama Affara
Department: Mathematics and Computer Science
University: Beirut Arab University

Topics Covered

Proof Methods
Recursion and Induction
Counting
Recurrence Relations
Algorithms
Graph Theory
Number Theory
Cryptography
Discrete Structures II

Graph Theory Outline

What are Graphs?
- Basic Terminology
- Graph Models
Representing Graphs
- Adjacency Lists
- Adjacency Matrices
- Incidence Matrices
Graph Theory
- Graph Isomorphism
- Graph Connectivity
- Euler and Hamilton Paths

Graph Definitions

Graph: A structure representing relationships
Vertices (Nodes): Connected by edges (arcs)

Formalizing Graphs

Specify vertices and edges
Vertices can be any item
- Persons
- Cities
- Neurons
- Webpages
Edges:
- Directed: ordered pair (u, v) from u to v
- Undirected: unordered pair {u, v}
Graph Definition: G=(V, E)
- V: set of vertices
- E: set of edges
Graph Models:
- Simple Graph
  - Undirected
  - No loops or multiple edges
- Multigraph
  - Undirected
  - Multiple edges are allowed
- Pseudograph
  - Undirected
  - Multiple edges and loops allowed
- Directed Graph
- Directed Multigraph
- Mixed Graph
- Graph with both directed and undirected edges
Graph Terminologies
- Self-loops: edge from a vertex to itself
- Multiple Edges: two or more edges connecting the same vertices

Adjacency

Vertices u and v are adjacent if there's an edge e={u, v}
The edge e is incident with u and v

Degree

Degree of Vertex (deg(v)): number of edges incident on v
Loop contributes twice
Indegree(deg−(v)): number of edges to v
Outdegree(deg+(v)): number of edges leaving v

Graph Theorem 1: Handshaking Theorem

2m = Σdeg(v)
- m: number of edges
- v: all vertices
Sum of degrees is twice the number of edges

Graph Theorem 2

Undirected graph has an even number of vertices with odd degrees

Graph Theorem 3

For directed graphs: |E| = Σ deg-(v) = Σ deg+(v)

Representing Graphs

Adjacency Lists: store vertices that are adjacent to each vertex
Adjacency Matrices
- n × n zero-one matrix
- (i,j)th entry is 1 if vertices vi and vj are adjacent; 0 otherwise.
- Symmetric for undirected graphs
Incidence Matrices
- n x m matrix
- (i,j)th entry is 1 if edge ej is incident with vertex vi; 0 otherwise.

Graph Isomorphism

Isomorphism: One-to-one, onto (bijective) function.
Preserves adjacency
Labelled Vertices: Number of graphs grows exponentially with the number of vertices
Unlabelled Vertices: Number of graphs is difficult to calculate

Graph Invariants

Properties preserved by isomorphism:
- Number of vertices (|V|)
- Number of edges (|E|)
- Minimal degree (d(G))
- Maximal degree (D(G))

Connectivity

Connected Graph: Path exists between every pair of distinct vertices
Connected Components: Disjoint connected subgraphs in a graph

Euler Paths and Circuits

Euler Path: Traverses each edge once
Euler Circuit: Starts and ends at the same vertex and traverses each edge once
Eulerian Graph: Contains an Euler circuit
Conditions for Eulerian Graphs:
- Connected
- All vertices have even degrees
Euler Path Conditions:
- Connected
- Exactly two vertices with odd degrees

Hamiltonian Paths and Circuits

Hamiltonian Path: Visits every vertex exactly once
Hamiltonian Cycle: Starts and ends at the same vertex and visits every vertex exactly once
Hamiltonian Graph: Contains a Hamiltonian cycle
No easy way to determine if a graph is Hamiltonian

Applications of Graph Isomorphism

Chemists/molecular graphs
- Model chemical compounds
- Vertices are atoms; edges are bonds
- Isomorphic to a known compound
Electronic circuits
- Model components and connections
- Verify circuit layouts
- Detect intellectual property theft

Applications of Euler Paths and Circuits

Neighborhood layout
Transportation networks
Utility connections
Communications networks

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Test your knowledge on fundamental concepts of graphs, including basic definitions, representations, and key theories such as isomorphism and connectivity. This quiz covers various proof methods, recursion, and algorithms related to graph theory. Ideal for students in Dr. Lama Affara's class at Beirut Arab University.