Abstract
The main goal of this paper is to estimate the magnitude of the second largest eigenvalue in absolute value, λ2, of (the adjacency matrix of) a randomd-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 thek-th step, for various values ofk. Our main theorem about eigenvalues is that
for any\(m \leqslant 2\left\lfloor {log n\left\lfloor {\sqrt {2d - } 1/2} \right\rfloor /\log d} \right\rfloor \), where E denotes the expected value over a certain probability space of 2d-regular graphs. It follows, for example, that for fixedd the second eigenvalue's magnitude is no more than\(2\sqrt {2d - 1} + 2\log d + C'\) with probability 1−n −C for constantsC andC′ for sufficiently largen.
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The author wishes to acknowledge the National Science Foundation for supporting this research in part under Grant CCR-8858788, and the Office of Naval Research under Grant N00014-87-K-0467.