. How many edges are there in a complete graph of 6 vertices? b. How many edges are there in a complete graph of 7 vertices? What is the difference in the number of edges in answers a and b. How do you explain it
The Correct Answer and Explanation is:
To calculate the number of edges in a complete graph, we use the formula for the number of edges in a complete graph with nnn vertices, which is:E=n(n−1)2E = \frac{n(n-1)}{2}E=2n(n−1)
Where EEE represents the number of edges, and nnn is the number of vertices.
a) For a complete graph with 6 vertices:
Using the formula:E=6(6−1)2=6×52=15E = \frac{6(6-1)}{2} = \frac{6 \times 5}{2} = 15E=26(6−1)=26×5=15
So, a complete graph with 6 vertices has 15 edges.
b) For a complete graph with 7 vertices:
Using the same formula:E=7(7−1)2=7×62=21E = \frac{7(7-1)}{2} = \frac{7 \times 6}{2} = 21E=27(7−1)=27×6=21
So, a complete graph with 7 vertices has 21 edges.
Difference in the number of edges:
Now, to find the difference between the number of edges in a complete graph with 7 vertices and one with 6 vertices:Difference=21−15=6\text{Difference} = 21 – 15 = 6Difference=21−15=6
Explanation:
The number of edges in a complete graph increases quadratically as the number of vertices increases. This is because each new vertex can connect to all the existing vertices, creating additional edges. Specifically, when you go from 6 vertices to 7 vertices, you are adding 6 more edges, one for each new connection between the new vertex and the 6 existing vertices. Hence, the difference of 6 edges reflects the number of new edges formed by adding an additional vertex to the graph.
In summary, as the number of vertices increases, the number of edges increases significantly, following the quadratic growth pattern governed by the formula n(n−1)2\frac{n(n-1)}{2}2n(n−1).
