By Giuliana Davidoff, Peter Sarnak, Alain Valette

This article is a self-contained research of expander graphs, in particular, their specific development. Expander graphs are hugely hooked up yet sparse, and whereas being of curiosity inside of combinatorics and graph idea, they could even be utilized to computing device technology and engineering. just a wisdom of undemanding algebra, research and combinatorics is needed as the authors give you the beneficial historical past from graph idea, quantity conception, staff thought and illustration idea. therefore the textual content can be utilized as a short creation to those matters and their synthesis in smooth arithmetic.

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**Additional resources for Elementary number theory, group theory, and Ramanujan graphs**

**Example text**

Check that, for 0 ≤ 2. 6, how large does n need to be taken in order to have g (X ) ≥ 10 and χ (X ) ≥ 10? N , 2 one has N N n . 7. 1. 1 can also be derived from the classical Perron–Frobenius theory. , the books by Biggs [5] and Chung [15]. 2. For treatments of families of expanders, see the books by Lubotzky [41] and Sarnak [57]. As indicated in the Overview, the construction of families of expanders is an important problem in network theory. The first constructions go back to the years 1972–73: using counting arguments, Pinsker [52] gave a nonexplicit construction, while Margulis [46] gave an explicit one by appealing to Kazhdan’s property (T ) in the representation theory of locally compact groups (see [17], Chapter 7, and [41], Chapter 3).

But r2 counts positive and negative solutions, while N2 counts only positive solutions; so N2 (r ) = d1 (r ) − d3 (r ). Now r ≡ 2 (mod. 4), so r2 is odd. Divisors of r which contribute to the latter formula are exactly divisors of r2 ; that is, N2 (r ) = (−1) a−1 2 ; (a,b):r =2ab similarly, N2 (s) = (−1) c−1 2 (c,d):s=2cd = (−1) 1−c 2 . ) Hence, N4 (4n) = (−1) a−c 2 . (a,b,c,d):4n=2ab+2cd a,b,c,d>0, odd Now we perform the change of variables a = x + y ; c = x − y ; b = z − t ; d = z + t, with inverse x= a+c a−c b+d d −b ; y= ; z= ; t= .

Proof of the Asymptotic Behavior Proof. Take L = √k k−1 ≥ 2 and ν = 1 n n−1 j=0 27 δ √µ j (where δa is the Dirac meak−1 sure at a ∈ [−L , L], that is, the probability measure on [−L , L] such that L −L f (x) d δa (x) = f (a), for every continuous function f on [−L , L]). 6. 8 are satisfied, and therefore there exists C = C(ε, k) > 0 such that ν [2 − ε, L] ≥ C. But µj 1 × (number of j’s with 2 − ε ≤ √ ≤ L) n k−1 √ 1 = × (number of eigenvalues of X in [(2 − ε) k − 1 , k]). 10. Theorem. Let (X m )m≥1 be a sequence of connected, k-regular, finite graphs for which g(X m ) → ∞ as m → ∞.