Number Of Simple Graph With n Vertices e Edges

Duration: 3 min

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AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video segment addresses a fundamental problem in graph theory: determining the total number of distinct simple graphs that can be formed with a fixed number of vertices (n) and a fixed number of edges (e). The instructor begins by defining the parameters for a specific example, setting the number of vertices to 4. He systematically derives the range of possible edges, identifying the minimum as zero and calculating the maximum using the standard formula for a complete graph. The core of the lesson involves applying combinatorial mathematics, specifically the combination formula, to select the required number of edges from the total pool of possible connections.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor writes the question 'Number of simple graph possible with n vertices and e edges' on the screen. He establishes the example parameters by writing |V| = 4. He explains that the minimum number of edges (E_min) is 0. He then derives the formula for the maximum number of edges (E_max) in a simple graph as n(n-1)/2. Substituting n=4, he calculates 4 * 3 / 2 = 6. He introduces the combination notation ^nC_r and its factorial formula n! / (r! (n-r)!), setting up the framework to solve for the number of graphs.

  2. 2:00 – 2:47 02:00-02:47

    The instructor proceeds to solve the specific problem for e=2 edges. He writes the combination expression ^6C_2, indicating the selection of 2 edges from the 6 possible edges calculated previously. He expands the factorials explicitly as 6! / (2! 4!). He simplifies the fraction by canceling out 4!, leaving (6 * 5) / 2. He performs the final arithmetic to get 15 and circles the result, concluding that there are 15 possible simple graphs with 4 vertices and 2 edges.

The lecture demonstrates a method for counting simple graphs by treating the set of all possible edges as a pool from which a specific subset must be chosen. By first calculating the total number of potential edges for a given number of vertices using the formula n(n-1)/2, the problem is reduced to a standard combinatorial selection. The instructor illustrates this by calculating the number of ways to choose 2 edges from the 6 available edges in a 4-vertex graph, resulting in 15 distinct graphs. This approach highlights the relationship between graph structure and combinatorial mathematics.

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