On the second eigenvalue and random walks n random d-regular graphs

The main goal of this paper is to estimate the magnitude of the second largest eigenvalue in absolute value, 2 , of (the adjacency matrix of) a random d-regular graph, G. In order to do so, we study the probability that a random walk on a random graph returns to its originating vertex at the k-th step, for various values of k. Our main theorem about eigenvalues is that E fj 2 (G)j m g 2 p 2d ? 1 1 + log d p 2d + O 1 p d ! + O d 3=2 loglog n logn ! m for any m 2 logn b p 2d ? 1=2c= logd , where E f g denotes the expected value over a certain probablity space of 2d-regular graphs. It follows, for example, that for xed d the second eigenvalue's magnitude is no more than 2 p 2d ? 1+2 log d+ C 0 with probability 1 ? n ?C for constants C and C 0 for suuciently large n.