Number Of Simple Graph With n Vertices

Duration: 5 min

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AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

The video lecture addresses the combinatorial problem of determining the total number of distinct simple graphs that can be formed with a given number of vertices, denoted as 'n'. The instructor begins by breaking down the problem for small values of n, specifically focusing on the case where n=3. He systematically enumerates all possible simple graphs by categorizing them according to the number of edges they contain, ranging from zero edges to the maximum possible edges. Through this enumeration, he demonstrates that for three vertices, there are exactly 8 possible simple graphs. The lecture then transitions to deriving a general formula for any number of vertices 'n' by considering the total number of potential edges in a complete graph. Finally, the instructor introduces a related but distinct problem: finding the number of simple graphs with 'n' vertices and a specific number of edges 'e'.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor poses the question "Number of simple graph possible with n vertices?" and begins a case-by-case analysis for n=3. He draws vertical columns to separate cases based on the number of edges, labeled |E|=0, |E|=1, |E|=2, and |E|=3. For |E|=0, he draws three isolated vertices. For |E|=1, he draws three distinct graphs, each with a single edge connecting a different pair of vertices. For |E|=2, he draws three graphs, each missing a different edge from the complete triangle. For |E|=3, he draws the complete graph K3 (a triangle). He sums the counts (1+3+3+1) to arrive at a total of 8 possible graphs for n=3.

  2. 2:00 – 4:37 02:00-04:37

    The instructor generalizes the previous result. He explains that the total number of graphs is 2 raised to the power of the number of possible edges. He writes the formula for the number of possible edges in a simple graph with n vertices as n(n-1)/2. Consequently, he writes the final formula for the total number of simple graphs as 2^(n(n-1)/2). He then moves to a new slide or question text: "Number of simple graph possible with n vertices and e edges". He underlines the terms "n vertices" and "e edges" to emphasize the constraints of this new problem, preparing to discuss combinations.

The lesson progresses from a concrete enumeration example to an abstract formula. By analyzing the n=3 case, the instructor illustrates that the number of graphs corresponds to the number of subsets of the set of all possible edges. Since each possible edge can either exist or not exist (2 choices), and there are n(n-1)/2 possible edges, the total count is 2^(n(n-1)/2). The video concludes by pivoting to a more constrained counting problem where the number of edges 'e' is fixed, shifting the focus from subsets of all possible edges to combinations of exactly 'e' edges chosen from the total possible edges.

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