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  3. exercise 4 aperiodicity of rw on graphs let xttz0 be...

Question: exercise 4 aperiodicity of rw on graphs let xttz0 be...

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Exercise 4 (Aperiodicity of RW on graphs). Let (Xt)tz0 be a random walk on a connected graph G (V,E) (i) Show that all nodes have the same period. (ii) If G contains an odd cycle C (e.g., triangle), show that all nodes in C have period 1. (iii) Show that Xt is aperiodic if and only if G contains an odd cycle. (iv)* (Optional) Show that X, is aperiodic if and only if G is bipartite. (A graph G is bipartite if there exists a partitionV-AuB of nodes such that there is no edge between A and B.)

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