13.1 Practice Question
Duration: 2 min
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This lecture segment focuses on solving practice problems in graph theory, specifically concerning the relationship between a simple graph G and its complement Gc. The instructor demonstrates how to calculate the number of vertices or edges using fundamental formulas involving complete graphs.
Chapters
0:00 – 2:00 00:00-02:00
The instructor introduces a problem where a simple graph G has 30 edges and its complement Gc has 36 edges, asking for the number of vertices n. The visible text displays the question 'Q A simple graph G has 30 edges and Gc has 36 edges, the number of vertices in G will be?'. The instructor writes the core formula |E(G)| + |E(Gc)| = n(n-1)/2, representing the total edges in a complete graph. Substituting 30 + 36 = n(n-1)/2, the instructor derives the quadratic equation n^2 - n = 132. The solution process involves factoring: n^2 - 12n + 11n - 132 = 0, leading to n(n-12) + 11(n-12) = 0, and finally (n+1)(n-12) = 0. The instructor notes x = -1 as a rejected solution, implying n=12 is the valid vertex count.
2:00 – 2:29 02:00-02:29
The second problem asks to find the number of edges in the complement |E(Gc)| given a simple graph G with |V|=8 and |E|=12. The instructor calculates the total possible edges for a complete graph with 8 vertices using the formula 8(8-1)/2, which equals 28. The visible calculation shows the subtraction step: 28 - 12, resulting in a final answer of 16. This value is circled on the screen as the solution for |E(Gc)|.
The lecture effectively demonstrates two distinct applications of the complement graph edge formula. The first example requires solving a quadratic equation to find the number of vertices n when given edge counts for both G and Gc. The second example is a direct arithmetic application where the total edges of a complete graph are calculated and the known edges of G are subtracted to find the complement's edge count. Both problems rely on the foundational concept that |E(G)| + |E(Gc)| equals n(n-1)/2.