How many edges has each of the following graphs: (a) K10; (b) K5,7; (c) W8; (d) the Petersen graph
The Correct Answer and Explanation is:
Let’s analyze each of the graphs and their number of edges:
(a) K10K_{10}K10 (Complete graph on 10 vertices):
In a complete graph KnK_nKn, every pair of distinct vertices is connected by an edge. The number of edges in KnK_nKn is given by the combination formula:E(Kn)=(n2)=n(n−1)2E(K_n) = \binom{n}{2} = \frac{n(n-1)}{2}E(Kn)=(2n)=2n(n−1)
For K10K_{10}K10, where n=10n = 10n=10:E(K10)=10(10−1)2=10×92=45E(K_{10}) = \frac{10(10-1)}{2} = \frac{10 \times 9}{2} = 45E(K10)=210(10−1)=210×9=45
So, K10K_{10}K10 has 45 edges.
(b) K5,7K_{5,7}K5,7 (Complete bipartite graph with partitions of size 5 and 7):
In a complete bipartite graph Km,nK_{m,n}Km,n, every vertex in the first set (size mmm) is connected to every vertex in the second set (size nnn), but there are no edges within each set. The number of edges is:E(Km,n)=m×nE(K_{m,n}) = m \times nE(Km,n)=m×n
For K5,7K_{5,7}K5,7, where m=5m = 5m=5 and n=7n = 7n=7:E(K5,7)=5×7=35E(K_{5,7}) = 5 \times 7 = 35E(K5,7)=5×7=35
Thus, K5,7K_{5,7}K5,7 has 35 edges.
(c) W8W_8W8 (Wheel graph with 8 vertices):
A wheel graph WnW_nWn is formed by taking a cycle of n−1n-1n−1 vertices and adding a central vertex connected to all others. The number of edges in WnW_nWn is given by:E(Wn)=n+(n−3)E(W_n) = n + (n-3)E(Wn)=n+(n−3)
where nnn is the total number of vertices, and the term n−3n-3n−3 comes from the connections to the center. For W8W_8W8, where n=8n = 8n=8:E(W8)=8+(8−3)=8+5=13E(W_8) = 8 + (8-3) = 8 + 5 = 13E(W8)=8+(8−3)=8+5=13
So, W8W_8W8 has 13 edges.
(d) Petersen graph:
The Petersen graph is a well-known graph with 10 vertices and 15 edges. It is often used as an example in graph theory due to its interesting properties (such as being a cubic graph, meaning each vertex has degree 3).
Thus, the Petersen graph has 15 edges.
Summary:
- (a) K10K_{10}K10: 45 edges
- (b) K5,7K_{5,7}K5,7: 35 edges
- (c) W8W_8W8: 13 edges
- (d) Petersen graph: 15 edges
