Podcast
Questions and Answers
In graph theory, what distinguishes a graph?
In graph theory, what distinguishes a graph?
- It represents relationships between objects using vertices and directed edges.
- It represents relationships between objects using vertices and edges. (correct)
- It is a data structure that can only store numerical data.
- It represents a singular object with multiple properties.
What condition must a set of vertices (V) satisfy for a structure to be considered a graph?
What condition must a set of vertices (V) satisfy for a structure to be considered a graph?
- V must contain only prime numbers.
- V must be finite and non-empty. (correct)
- V must be empty.
- V must contain an infinite number of vertices.
If two graphs, G and H, are considered equal, what characteristic must they share?
If two graphs, G and H, are considered equal, what characteristic must they share?
- Same number of vertices but different edge arrangements
- Same order of vertices but different edge arrangements
- Different sets of vertices but interchangeable edges
- Identical sets of vertices and identical sets of edges. (correct)
Considering adjacency list representation, what information is primarily stored for each vertex in a graph?
Considering adjacency list representation, what information is primarily stored for each vertex in a graph?
In the context of graph data structures, how is an adjacency matrix A defined for a graph with V vertices?
In the context of graph data structures, how is an adjacency matrix A defined for a graph with V vertices?
What key feature defines a null graph?
What key feature defines a null graph?
What distinguishes a simple graph from other types of graphs?
What distinguishes a simple graph from other types of graphs?
What is true of edges in an undirected graph?
What is true of edges in an undirected graph?
What is the alternative term for a directed graph?
What is the alternative term for a directed graph?
Which characteristic defines a connected graph?
Which characteristic defines a connected graph?
In a disconnected graph, what is a fundamental property regarding paths between vertices?
In a disconnected graph, what is a fundamental property regarding paths between vertices?
What conditions are necessary for a graph with 'n' vertices to be considered cyclic?
What conditions are necessary for a graph with 'n' vertices to be considered cyclic?
What does the distance between two vertices represent in a graph?
What does the distance between two vertices represent in a graph?
What does the eccentricity of a vertex measure?
What does the eccentricity of a vertex measure?
How is the radius of a connected graph defined?
How is the radius of a connected graph defined?
What is the diameter of a graph?
What is the diameter of a graph?
Under what condition is a vertex considered a central point in a graph?
Under what condition is a vertex considered a central point in a graph?
According to the sum of degrees of vertices theorem, what is the relationship between the sum of the degrees of all vertices and the number of edges in a non-directed graph?
According to the sum of degrees of vertices theorem, what is the relationship between the sum of the degrees of all vertices and the number of edges in a non-directed graph?
What does it mean for each element of E to be a 2-element subset of V?
What does it mean for each element of E to be a 2-element subset of V?
If a graph has vertices V = {A, B, C, D} and the edges E = {{A,B}, {B,C}, {C,D}, {D,A}}, what type of graph is it?
If a graph has vertices V = {A, B, C, D} and the edges E = {{A,B}, {B,C}, {C,D}, {D,A}}, what type of graph is it?
Flashcards
What is a graph?
What is a graph?
A structure representing relationships between objects, with vertices representing objects and edges representing connections.
Adjacency List/Dictionary
Adjacency List/Dictionary
Storing a graph using a dictionary or list, where each vertex is associated with its neighboring vertices.
Adjacency Matrix
Adjacency Matrix
Storing a graph as a matrix where rows and columns represent vertices, and entries indicate the presence or absence of edges.
Null Graph
Null Graph
Signup and view all the flashcards
Simple Graph
Simple Graph
Signup and view all the flashcards
Undirected Graph
Undirected Graph
Signup and view all the flashcards
Directed Graph
Directed Graph
Signup and view all the flashcards
Connected Graph
Connected Graph
Signup and view all the flashcards
Disconnected Graph
Disconnected Graph
Signup and view all the flashcards
Cyclic Graph
Cyclic Graph
Signup and view all the flashcards
Distance Between Vertices
Distance Between Vertices
Signup and view all the flashcards
Eccentricity of a Vertex
Eccentricity of a Vertex
Signup and view all the flashcards
Radius of a Connected Graph
Radius of a Connected Graph
Signup and view all the flashcards
Diameter of a Graph
Diameter of a Graph
Signup and view all the flashcards
Central Point
Central Point
Signup and view all the flashcards
Sum of Degrees Theorem
Sum of Degrees Theorem
Signup and view all the flashcards
Study Notes
- Graph theory is a structure that represents relationships between objects.
- Vertices (V) represent objects, and edges (E) represent connections between objects.
- A graph (G) is an ordered pair of two sets: vertex set (V) and edge set (E).
- V must be finite and non-empty.
- Each element of E is a 2-element subset of V.
- Therefore, G = (V, E).
Example Graph
- Vertex set V = {1, 2, 3, 4}.
- Edge set E = {{1,2}, {2,3}, {3,4}, {1,4}, {2,4}, {1,3}}.
- G = ({1, 2, 3, 4}, {{1,2}, {2,3}, {3,4}, {1,4}, {2,4}, {1,3}}).
- Two graphs, G and H, are equal only if they have the same set of vertices and edges.
Data Structures for Storing Graphs
- Graphs are stored as an adjacency list or dictionary.
- The adjacency list or dictionary stores each vertex's neighbor.
- Adjacency Matrix: Matrix A of dimensions V x V is defined
Types of Graphs
- Null Graph: Has no edges between vertices, also called an empty graph.
- Simple Graph: Undirected graph with no parallel edges or loops; for 'n' vertices, the degree of any vertex is at most n-1.
- Undirected Graph: A graph where edges are not directed.
- Directed Graph: A graph whose edges are directed by arrows, also known as a digraph.
- Connected Graph: It is possible to visit any vertex from any other vertex; at least one edge or path exists between every pair of vertices.
- Disconnected Graph: No path exists between every pair of vertices.
- Cyclic Graph: 'n' vertices (where n >= 3) and 'n' edges forming a cycle, each vertex has a degree of 2.
Basic Properties of Graphs
- Distance Between Two Vertices: The number of edges in the shortest path between two vertices.
- Eccentricity of a Vertex: The maximum distance from that vertex to all other vertices.
- Radius of a Connected Graph: The minimum eccentricity from all vertices.
- Diameter of the Graph: The maximum eccentricity from all vertices.
- Central Point: If the eccentricity of a graph equals its radius.
- Sum of the Degrees of Vertices Theorem: For a non-directed graph G = (V, E) where Vertex set V = {V1, V2, ...., Vn}, the sum of degree of all the vertices equals twice the number of edges in the graph.
Summary Equation
- ∑ deg(Vi) = 2E
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
