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We assume the connection between the cluster heads form a complete graph and the graph is completely connected.
If every pair of vertices is connected by an edge, the graph is called a complete graph (Figure 13B).
Fig. 1 Decomposition of a complete graph.
Now suppose further that is a complete graph.
So ({mathbb {G}}(A cong K_2) is a complete graph.
Thus ({mathbb {G}}(A)) is a complete graph.
Conversely, suppose that ({mathbb {G}}(A)) is a complete graph.
We use the term metric graph for a complete graph with metric weights.
We consider the p-lazy simple random walk on a complete graph with N+1-vertices.
Now, Theorem 9 implies that ({mathbb {G}}(A)) is a complete graph.
For the other side assume that ({mathbb {G}}({mathbb {Z}}_n)) is a complete graph.
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