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Additional resources for Elementary number theory, group theory, and Ramanujan graphs
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  and Chung . 2. For treatments of families of expanders, see the books by Lubotzky  and Sarnak . 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  gave a nonexplicit construction, while Margulis  gave an explicit one by appealing to Kazhdan’s property (T ) in the representation theory of locally compact groups (see , Chapter 7, and , 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 → ∞.