A d-regular graph has largest or first (adjacency matrix) eigenvalue λ 1 = d. Consider for an even d ≥ 4, a random d-regular graph model formed from d/2 uniform, independent permutations on {1, . . . , n}. We shall show that for any ǫ > 0 we have all eigenvalues aside fromWe also show that this probability is at most 1 − c/n τ ′ , for a constant c and a τ ′ that is either τ or τ + 1 ("more often" τ than τ + 1). We prove related theorems for other models of random graphs, including models with d odd.These theorems resolve the conjecture of Alon, that says that for any ǫ > 0 and d, the second largest eigenvalue of "most" random dregular graphs are at most 2 √ d − 1 +ǫ (Alon did not specify precisely what "most" should mean or what model of random graph one should take).

The main goal of this paper is to estimate the magnitude of the second largest eigenvalue in absolute value, A2, 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 thatfor any m < 2[logn L2dv/~g~-l/2J/logdJ, where E denotes the expected value over a certain probability space of 2d-regular graphs. It follows, for example, that for fixed d the second eigenvalue's magnitude is no more than 2 2~/'2d "L--1 + 2 log d + C I with probability 1 -n -C for constants C and C' for sufficiently large n. j fecs.princeton, edu

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We study three mathematical notions, that of nodal regions for eigenfunctions of the Laplacian, that of covering theory, and that of fiber products, in the context of graph theory and spectral theory for graphs. We formulate analogous notions and theorems for graphs and their eigenpairs. These techniques suggest new ways of studying problems related to spectral theory of graphs. We also perform some numerical experiments suggesting that the fiber product can yield graphs with small second eigenvalue.

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