## Abstract

The main goal of this paper is to estimate the magnitude of the second largest eigenvalue in absolute value, λ_{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

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 2*d*-regular graphs. It follows, for example, that for fixed*d* the second eigenvalue's magnitude is no more than\(2\sqrt {2d - 1} + 2\log d + C'\) with probability 1−*n*
^{−C} for constants*C* and*C*′ for sufficiently large*n*.

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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 random*d*-regular graphs.
*Combinatorica* **11, **331–362 (1991). https://doi.org/10.1007/BF01275669

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### AMS subject classification code (1991)

- 05 C 50
- 05 C 80