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Questions and Answers
What does the in-degree of a vertex represent in a directed graph?
What does the in-degree of a vertex represent in a directed graph?
In the directed graph defined by edges E = {(A, B), (A, C), (B, C), (C, D)}, what is the out-degree of vertex A?
In the directed graph defined by edges E = {(A, B), (A, C), (B, C), (C, D)}, what is the out-degree of vertex A?
What is the handshaking theorem primarily used for?
What is the handshaking theorem primarily used for?
What does the maximum degree represent in a graph?
What does the maximum degree represent in a graph?
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Given a directed graph, if vertex D has an in-degree of 2, how many edges must be connected to D?
Given a directed graph, if vertex D has an in-degree of 2, how many edges must be connected to D?
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If the degree sequence of a directed graph is given as {4, 3, 2, 1}, which statement is true?
If the degree sequence of a directed graph is given as {4, 3, 2, 1}, which statement is true?
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How is the degree sequence arranged for a graph?
How is the degree sequence arranged for a graph?
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Which of the following statements is NOT true about directed graphs?
Which of the following statements is NOT true about directed graphs?
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What feature characterizes a directed graph?
What feature characterizes a directed graph?
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What is the maximum number of edges allowed between two vertices in a simple directed graph?
What is the maximum number of edges allowed between two vertices in a simple directed graph?
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Which statement about loops in directed graphs is true?
Which statement about loops in directed graphs is true?
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In a directed graph with edge multiplicity, what does it mean if an edge has multiplicity m?
In a directed graph with edge multiplicity, what does it mean if an edge has multiplicity m?
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What distinguishes undirected graphs from directed graphs?
What distinguishes undirected graphs from directed graphs?
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Which condition is NOT present in a simple directed graph?
Which condition is NOT present in a simple directed graph?
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In a directed graph with the vertices A, B, C, and D, if there is an edge from A to B and another from B to A, what type of edge is this?
In a directed graph with the vertices A, B, C, and D, if there is an edge from A to B and another from B to A, what type of edge is this?
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If a graph contains vertices A, B, and C where A has edges to both B and C, what type of graph does this describe?
If a graph contains vertices A, B, and C where A has edges to both B and C, what type of graph does this describe?
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What is the term used for two vertices that share an edge in an undirected graph?
What is the term used for two vertices that share an edge in an undirected graph?
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In an undirected graph, how does a loop affect the degree of a vertex?
In an undirected graph, how does a loop affect the degree of a vertex?
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If vertex A has edges (A, B) and (A, D), what is the degree of vertex A in an undirected graph?
If vertex A has edges (A, B) and (A, D), what is the degree of vertex A in an undirected graph?
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In a directed graph, if there is an edge (u, v), how are u and v described in relation to each other?
In a directed graph, if there is an edge (u, v), how are u and v described in relation to each other?
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What is the correct way to denote the degree of a vertex v in a graph?
What is the correct way to denote the degree of a vertex v in a graph?
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Which statement is true regarding edges in an undirected graph?
Which statement is true regarding edges in an undirected graph?
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If a graph consists of the vertices A, B, C, and D with edges {(A, B), (B, C), (C, D)}, how many edges are considered as incident to vertex B?
If a graph consists of the vertices A, B, C, and D with edges {(A, B), (B, C), (C, D)}, how many edges are considered as incident to vertex B?
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Which of the following describes a scenario where vertices A and C are adjacent in a directed graph?
Which of the following describes a scenario where vertices A and C are adjacent in a directed graph?
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Study Notes
Basic Terminology
- Degree of a vertex in an undirected graph counts the number of edges incident to it; loops at the vertex contribute twice to the degree.
- Denoted as deg(v) for a vertex v.
Vertex Degree Examples
- In a graph with vertices P, Q, R, and S:
- deg(P) = 2
- deg(Q) = 3
- deg(R) = 2
- deg(S) = 1
Adjacent Vertices
- Two vertices u and v are adjacent if they share an edge e.
- An edge e connects u and v; u is the initial vertex and v is the terminal vertex.
- For loops, the initial and terminal vertices are the same.
Directed Graphs
- In directed graphs, if there is an edge (u, v), then u is adjacent to v and v is adjacent from u.
Edge Multiplicity
- If m different edges connect the same unordered pair of vertices {u, v}, it's referred to as an edge of multiplicity m.
Self-Connecting Edges
- Loops represent edges connecting a vertex to itself, important for modeling networks, e.g. communications links.
Directed Graph Characteristics
- Directed graphs consist of vertices and directed edges (arcs) with arrows indicating the direction.
- Can contain loops, multiple edges, and bidirectional edges.
Simple Directed Graph
- A simple directed graph has no loops or multiple edges; at most one edge exists between any two vertices.
- An edge of multiplicity m is considered if m directed edges exist between vertices (u, v).
In-Degree and Out-Degree
- In a directed graph, in-degree (deg⁻(v)) counts edges leading to vertex v, while out-degree (deg⁺(v)) counts edges originating from vertex v.
- Loops contribute 1 to both in-degree and out-degree of the vertex.
Degree Sequence
- The degree sequence arranges vertex degrees in non-increasing order.
- In directed graphs, both in-degrees and out-degrees are considered.
Example of Degree Sequence
- For the graph G with vertices {A, B, C, D} and edges {(A, B), (A, C), (B, C), (C, D)}:
- Degree sequence in non-increasing order: {3, 2, 2, 1}
- Minimum degree is the smallest degree of any vertex; maximum degree is the largest.
Theorems
- For directed graphs, the total degree sum equals the number of directed edges:
- ∑ deg(v) = |E|
- Handshaking Theorem for undirected graphs states the sum of all vertex degrees equals twice the number of edges:
- ∑ deg(v) = 2m, applicable even with multiple edges and loops.
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Description
This quiz covers the basic terminology of graph theory, focusing on key concepts like the degree of a vertex and its significance in undirected graphs. Test your knowledge of how relationships in a graph can be represented and analyzed. Perfect for those studying introductory graph theory concepts.
