## Abstract

We study here the spectra of random lifts of graphs. Let G be a finite connected graph, and let the infinite tree T be its universal cover space. If λ_{1} and ρ are the spectral radii of G and T respectively, then, as shown by Friedman (Graphs Duke Math J 118 (2003), 19-35), in almost every n-lift H of G, all "new" eigenvalues of H are ≤ O(λ^{1/2}_{1} ρ ^{1/2}). Here we improve this bound to O(λ^{1/3}_{1} ρ^{2/3}). It is conjectured in (Friedman, Graphs Duke Math J 118 (2003) 19-35) that the statement holds with the bound ρ +o(1) which, if true, is tight by (Greenberg, PhD thesis, 1995). For G a bouquet with d/2 loops, our arguments yield a simple proof that almost every d-regular graph has second eigenvalue O(d^{2/3}). For the bouquet, Friedman (2008). has famously proved the (nearly?) optimal bound of 2 √ d-1 +o(1). Central to our work is a new analysis of formal words. Let w be a formal word in letters g^{± 1},...,g^{± 1}. The word map associated with w maps the permutations σ_{1},...,σ_{k} ∈ S_{n} to the permutation obtained by replacing for each i, every occurrence of g_{i} in w by σ_{i}. We investigate the random variable X^{n}_{w} that counts the fixed points in this permutation when the σ_{i} are selected uniformly at random. The analysis of the expectation E(X^{n}_{w}) suggests a categorization of formal words which considerably extends the dichotomy of primitive vs. imprimitive words. A major ingredient of a our work is a second categorization of formal words with the same property. We establish some results and make a few conjectures about the relation between the two categorizations. These conjectures suggest a possible approach to (a slightly weaker version of) Friedman's conjecture. As an aside, we obtain a new conceptual and relatively simple proof of a theorem of A. Nica (Nica, Random Struct Algorithms 5 (1994), 703-730), which determines, for every fixed w, the limit distribution (as n → ∞) of X^{n}_{w}. A surprising aspect of this theorem is that the answer depends only on the largest integer d so that w = u^{d} for some word u.

Original language | English |
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Pages (from-to) | 100-135 |

Number of pages | 36 |

Journal | Random Structures and Algorithms |

Volume | 37 |

Issue number | 1 |

DOIs | |

State | Published - Aug 2010 |

Externally published | Yes |

## Keywords

- Alon's conjecture
- Graph lifts
- Random graphs
- Spectrum of graphs
- Word map