How to find the number of edges of $K_{n_1,n_2,...,n_m}$ ?
it is easy to find the number of edge of $K_{n_1}$ , but how if i want to find the number of edges of $K_{n_1,n_2,...,n_m}$ ?
2026-03-29 16:39:52.1774802392
How to find the number of edges of $K_{n_1,n_2,...,n_m}$
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Start by making it a complete graph by adding edges within each component of this multipartite graph.
Notice the number of edges you added is $\sum_{i=1}^m\frac{n_i(n_i-1)}{2}$.
The total number of edges now is $\frac{\sum_{i=1}^mn_i(\sum_{i=1}^mn_i-1)}{2}$ as this now a complete graph.
Taking the difference, this is the number of edges originally in the graph:
$\frac{\sum_{i=1}^mn_i(\sum_{i=1}^mn_i-1)}{2}-\sum_{i=1}^m\frac{n_i(n_i-1)}{2}$.