Math Problem Statement

Tentukan diketahui sebuah graf berarah E dengan himpunan V={v1, v2, v3, v4, v5} dan himpunan arah E={(v1, v2) (v2, v3) (v3, v4) (v4, v1) (v5, v5) (v5, v1)}gambar lah diagram dari graf E

Tentukan diketahui sebuah graf berarah E dengan himpunan V = {v1, v2, v3, v4, v5} dan himpunan arah E = {(v1, v2), (v2, v3), (v3, v4), (v4, v1), (v5, v5), (v5, v1)} gambarlah diagram dari graf E

Solution

The problem asks you to draw a directed graph (or digraph) EE based on the given sets:

  1. Vertices (V):
    V={v1,v2,v3,v4,v5}V = \{v_1, v_2, v_3, v_4, v_5\}

  2. Edges (E):
    E={(v1,v2),(v2,v3),(v3,v4),(v4,v1),(v5,v5),(v5,v1)}E = \{(v_1, v_2), (v_2, v_3), (v_3, v_4), (v_4, v_1), (v_5, v_5), (v_5, v_1)\}

Steps to draw the directed graph:

  1. Vertices: Place 5 vertices labeled v1,v2,v3,v4,v5v_1, v_2, v_3, v_4, v_5.
  2. Edges: Connect the vertices with directed edges (arrows) based on the following pairs:
    • (v1,v2)(v_1, v_2): Draw an arrow from v1v_1 to v2v_2.
    • (v2,v3)(v_2, v_3): Draw an arrow from v2v_2 to v3v_3.
    • (v3,v4)(v_3, v_4): Draw an arrow from v3v_3 to v4v_4.
    • (v4,v1)(v_4, v_1): Draw an arrow from v4v_4 to v1v_1, forming a cycle with v1,v2,v3,v4v_1, v_2, v_3, v_4.
    • (v5,v5)(v_5, v_5): Draw a loop from v5v_5 to itself.
    • (v5,v1)(v_5, v_1): Draw an arrow from v5v_5 to v1v_1.

This will result in a directed graph where:

  • There is a cycle connecting v1,v2,v3,v4v_1, v_2, v_3, v_4,
  • v5v_5 has a self-loop and is connected to v1v_1.

Would you like a detailed image of the graph or further clarification?


Here are 5 related questions to expand your understanding:

  1. What is the difference between directed and undirected graphs?
  2. How do you identify cycles in directed graphs?
  3. What are the in-degrees and out-degrees of each vertex in this graph?
  4. What are the practical applications of directed graphs?
  5. How do self-loops affect the structure of a graph?

Tip: In a directed graph, the direction of edges is crucial for understanding relationships between vertices, unlike in an undirected graph.

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Math Problem Analysis

Mathematical Concepts

Graph Theory
Directed Graphs
Vertices
Edges
Cycles

Formulas

-

Theorems

Graph Cycles
Self-loops in Directed Graphs

Suitable Grade Level

University Level