# On the second eigenvalue and random walks in randomd-regular graphs

## 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

$$E|\lambda _2 (G)|^m \leqslant \left( {2\sqrt {2d - 1} \left( {1 + \frac{{\log d}}{{\sqrt {2d} }} + 0\left( {\frac{1}{{\sqrt d }}} \right)} \right) + 0\left( {\frac{{d^{3/2} \log \log n}}{{\log n}}} \right)} \right)^m$$

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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## Additional information

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.

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### Cite this article

Friedman, J. On the second eigenvalue and random walks in randomd-regular graphs. Combinatorica 11, 331–362 (1991). https://doi.org/10.1007/BF01275669

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