vi, 48 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We compare the relaxation times of two random walks - the simple random walk and the metropolis walk - on an arbitrary finite multigraph G. We apply this result to the random graph with n vertices, where each edge is included with probability p = [Special characters omitted.] where λ > 1 is a constant and also to the Newman-Watts small world model. We give a bound for the reconstruction problem for general trees and general 2 × 2 matrices in terms of the branching number of the tree and some function of the matrix. Specifically, if the transition probabilities between the two states in the state space are a and b , we show that we do not have reconstruction if Br( T ) [straight theta] < 1, where [Special characters omitted.] and Br( T ) is the branching number of the tree in question. This bound agrees with a result obtained by Martin for regular trees and is obtained by more elementary methods. We prove an inequality closely related to this problem. / Committee in charge: David Levin, Chairperson, Mathematics; Christopher Sinclair, Member, Mathematics; Marcin Bownik, Member, Mathematics; Hao Wang, Member, Mathematics; Van Kolpin, Outside Member, Economics