Graph Theory Fundamentals
18 Questions
4 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the definition of a vertex in the context of graph theory?

  • A data structure representing a set of edges.
  • A labeled object in a graph. (correct)
  • A connection between two vertices.
  • A numerical value assigned to edges.

    • Which of the following graph representations explicitly lists vertices and edges?

  • Adjacency Matrix
  • Adjacency List
  • Incidence Matrix
  • Vertex and Edge List (correct)

    • In a complete graph denoted as K_n, what is true about the vertices?

  • Every vertex connects to a central vertex.
  • No edges connect within the set of vertices.
  • Vertices are arranged in a circle without edges.
  • Every vertex connects to every other vertex. (correct)

    • Which theorem states that the sum of the degrees of all vertices equals twice the number of edges?

    <p>Handshaking Lemma</p> Signup and view all the answers

    Which type of graph has a path that visits every edge exactly once and can return to the starting vertex?

    <p>Eulerian Circuit</p> Signup and view all the answers

    What characteristic defines a tree in graph theory?

    <p>A connected acyclic graph with edges equal to vertices minus one.</p> Signup and view all the answers

    What does the chromatic number of a graph represent?

    <p>The minimum number of colors needed for proper vertex coloring.</p> Signup and view all the answers

    What is the primary function of Dijkstra’s Algorithm in graph theory?

    <p>To find the shortest path between two vertices in a weighted graph.</p> Signup and view all the answers

    What does the neighborhood of a vertex in a graph represent?

    <p>The set of vertices adjacent to the vertex</p> Signup and view all the answers

    Which of the following statements about Hamiltonian paths is correct?

    <p>They visit every vertex exactly once.</p> Signup and view all the answers

    How does a complete graph differ from a bipartite graph?

    <p>Bipartite graphs have edges only between two distinct sets of vertices.</p> Signup and view all the answers

    What is a distinguishing feature of Dijkstra’s Algorithm?

    <p>It finds the shortest path in a weighted graph.</p> Signup and view all the answers

    In the context of graph theory, what defines an Eulerian circuit?

    <p>It returns to the starting vertex only if all edges are visited.</p> Signup and view all the answers

    What information does an incidence matrix convey in graph theory?

    <p>The connections between vertices and edges in a graph.</p> Signup and view all the answers

    If a tree has 'n' vertices, how many edges does it contain?

    <p>n-1</p> Signup and view all the answers

    What does the term 'graph isomorphism' refer to?

    <p>Graphs that can be transformed into one another without changing the connection structure.</p> Signup and view all the answers

    Which of the following is NOT a characteristic of a cycle graph?

    <p>It can have vertices connected to other vertices outside the cycle.</p> Signup and view all the answers

    Which statement correctly describes the degree of a vertex in a graph?

    <p>It indicates how many edges are connected to the vertex.</p> Signup and view all the answers

    Study Notes

    Graph Theory Fundamentals

    • Vertices and Edges: A graph comprises vertices (nodes) – labeled objects – and edges (connections) – links or arcs. Vertices are labeled objects; edges connect them.

    • Degree of a Vertex: The degree (d(v)) of a vertex is the number of edges incident to it.

    • Adjacency: Adjacent vertices are those connected by an edge. Their set of neighbors (N(v)) is the adjacent vertices to v.

    Graph Representations

    • Vertex/Edge List: Explicitly lists all vertices and edges.

    • Adjacency List: Each vertex has a list of its adjacent vertices.

    • Adjacency Matrix: A matrix where rows and columns represent vertices; entries indicate the presence of edges.

    • Incidence Matrix: A matrix showing the connections between vertices and edges.

    Graph Types

    • Complete Graph (Kn): Every vertex is connected to every other.

    • Cycle Graph (Cn): Vertices form a closed loop.

    • Star Graph (Sn): A central vertex connects to all others.

    • Bipartite Graph: Vertices are divided into two sets, with edges only joining vertices across sets.

    Key Theorems

    • Handshaking Lemma: The sum of vertex degrees equals twice the number of edges ( Σd(v) = 2|E| ).

    • Isomorphism: Equivalent graphs with a one-to-one correspondence between vertices preserving adjacency.

    Paths and Cycles

    • Path: A sequence of edges connecting vertices.

    • Eulerian Path/Circuit: A path/circuit visiting every edge exactly once.

    • Hamiltonian Path/Circuit: Visits every vertex exactly once; Hamiltonian Circuit returns to the starting vertex.

    Trees

    • Trees: Connected, acyclic graphs. A tree with n vertices has n-1 edges.

    Graph Coloring

    • Graph Coloring: Assigns colors to vertices so no adjacent vertices have the same color.

    • Chromatic Number (χ(G)): The minimum number of colors needed to color the graph.

    Algorithms

    • Dijkstra's Algorithm: Finds shortest paths in weighted graphs from a starting vertex.

    Studying That Suits You

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

    Quiz Team

    Description

    This quiz covers the fundamental concepts of graph theory, including vertices, edges, and various graph representations. Test your understanding of different graph types such as complete graphs, cycle graphs, and bipartite graphs. Challenge yourself to identify and differentiate these key elements.

    More Like This

    Use Quizgecko on...
    Browser
    Open
    Browser
    Browser