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547Hitting Time on K₄ Minus One EdgeTake the complete graph K 4 on vertices \ 1,2,3,4\ and remove edge \ 1,4\ . The resulting graph has 5 edges, with d(1)=d(4)=2 and d(2)=d(3)=3. A simple random walk moves at each step to a uniformly random neighbor. Starting from vertex 2, what is the expected number of steps to reach vertex 4?概率简单数值题未尝试免费548Commute Time Between Endpoints of a PathConsider the path graph P n with vertices \ 0, 1, \ldots, n-1\ and n-1 edges (each of unit resistance), where the walk at interior vertices moves left or right with equal probability, and at the endpoints moves to the unique neighbor. (a) Compute the effective resistance R eff (0, n-1) between the two endpoints. (b) Using C(u,v) = 2m R eff (u,v), find the commute time between the two endpoints. (c) Verify for n = 4 by computing h(0 3) and h(3 0) directly.概率中等derivation未尝试免费550Expected Cover Time of the Cycle C₆A simple random walk moves on the cycle graph C 6 (vertices 0, 1, \ldots, 5). At each step, the walker moves clockwise or counterclockwise with equal probability. Starting at vertex 0, what is the expected number of steps to visit all 6 vertices (the expected cover time)?概率困难derivation未尝试面试订阅