DIAMETER OF RAMANUJAN GRAPHS AND RANDOM CAYLEY GRAPHS

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1 Bolyai Society Springer-Verlag Combinatorica 1 20 DIAMETER OF RAMANUJAN GRAPHS AND RANDOM CAYLEY GRAPHS NASER T. SARDARI Received March 2, 2016 Revised September 2, 2017 We study the diameter of LPS Ramanujan graphs X p,q. We show that the diameter of the bipartite Ramanujan graphs is greater than (/3)log p (n)+o(1), where n is the number of vertices of X p,q. We also construct an infinite family of (p + 1)-regular LPS Ramanujan graphs X p,m such that the diameter of these graphs is greater than or equal to b(/3)log p (n)c. On the other hand, for any k-regular Ramanujan graph we show that the distance of only a tiny fraction of all pairs of vertices is greater than (1+ )log k 1 (n). We also have some numerical experiments for LPS Ramanujan graphs and random Cayley graphs which suggest that the diameters are asymptotically (/3)log k 1 (n) andlog k 1 (n), respectively. 1. Introduction 1.1. Motivation The diameter of any k-regular graph with n vertices is bounded from below by log k 1 (n) and it could get as large as a scalar multiple of n. Itisknown that the diameter of any k-regular Ramanujan graph is bounded from above by 2(1 + )log k 1 (n) [6]. Lubotzky, Phillips and Sarnak constructed an explicit family of (p + 1) regular Ramanujan graphs X p,q [6], where p and q are prime numbers and q 1 mod. X p,q is a p+1-regular bipartite or nonbipartite graph depending on p being a non-quadratic or quadratic residue modulo q, respectively. Their construction can be modified for every integer q and prime p 3 mod ; see [3], [] or [7]. It was expected that the diameter of the LPS Ramanujan graphs to be bounded from above by (1+ )log k 1 (n); Mathematics Subject Classification (2010):... Fill in, please

2 2 NASER T. SARDARI see [10, Chapter 3]. However, we show that the diameter of an infinite family of p+1-regular LPS Ramanujan graphs is greater than or equal to (1.1) log 3 p (n). While there are points x and y whose distance is large in a LPS Ramanujan graph, we prove that the distance of a tiny fraction of vertices in any k-regular Ramanujan graph G is less than (1+ )log k 1 (n). In other words, the essential diameter is asymptotic to (1 + ) log k 1 (n), where the essential diameter of a graph is d if 99% of the distance of pairs of vertices is less than d. In fact, we prove a stronger result, we show that for every vertex x in a k-regular Ramanujan graph G the number of points which cannot be visited by exactly l steps, where l>(1+ )log k 1 (n), is less than n 1. So the density of them is O(n ). In particular, it also recovers 2(1+ )log k 1 (n) as an upper bound on the diameter of k-regular Ramanujan graph. Furthermore, we give some numerical results for two families of 6-regular graphs. The first family of graphs are the 6-regular LPS Ramanujan graphs and we denote them by X 5,q. The second family are the 6-regular random Cayley graphs PSL 2 (Z/qZ), i.e., the Cayley graphs that are constructed by 3 random generators of PSL 2 (Z/qZ) and their inverses {s ± 1,s± 2,s± 3 }. We denote these graphs by Zq. The numerical experiments suggest that the diameter of LPS Ramanujan graphs is asymptotic to (1.2) log 3 5 (n). This is consistent with our conjecture on the optimal strong approximation for quadratic forms in variables [8]. On the other hand, the numerical data suggests that the diameter of the random Cayley graph equals that of a random 6-regular graph [2], that is (1.3) log 5 (n). The archimedean analog of our question has been discussed in Sarnak s letter to Scott Aaronson and Andy Pollington; see [11]. In that context, the approximation of points on the sphere by words in LPS generators is considered. This question is related to the theory of quadratic Diophantine equations; see [8]. Sarnak defines the notion of the covering exponent and the almost all covering exponent [11, Page 3] that are the analogue of diameter and the essential diameter in our paper. Sarnak showed that the almost all

3 RAMANUJAN GRAPHS AND RANDOM CAYLEY GRAPHS 3 covering exponent is 1; see [11, Page 28]. Our Theorem 1.5 is the p-adic analogue of Sarnak s theorem. In a recent paper [5], Lubetzky and Peres show the simple random walk exhibits cuto on Ramanujan graphs. As a result they give a more detailed version of our Theorem 1.5. In a similar work, for the family of LPS bipartite Ramanujan graphs, Biggs and Boshier determined the asymptotic behavior of the girth of these graphs; see [1]. They showed that the girth is asymptotic to 3 log k 1 (n) Statement of results We begin by a brief description of LPS Ramanujan graphs; see [6] for a comprehensive treatment of them. The idea of the construction is coming from number theory, i.e., generalized Ramanujan conjecture. More precisely, we consider the symmetric space PGL 2 (Q p )/PGL 2 (Z p ) which can be identified with a regular (p + 1)-infinite tree and form the double coset \PGL 2 (Q p )/PGL 2 (Z p ), where is a suitable arithmetic discrete subgroup of PGL 2 (Q p ). In LPS Ramanujan graphs, the authors take =H (Z[1/p]), where H is the unit group of Hamiltonian quaternion. We note that H splits at any prime p 6= 2 and therefore, H (Z[1/p]) acts from the left on PGL 2 (Q p )/PGL 2 (Z p ). The generalized Ramanujan conjecture, which is a theorem for Hamilton quaternion H, implies that the quotient of PGL 2 (Q p )/PGL 2 (Z p ) by any congruence subgroup of H (Z[1/p]) which is afinitep+1-regular graph is a Ramanujan graph. By considering an appropriate congruence subgroup of H (Z[1/p]) we can identify the quotient of this symmetric space with a Cayley graph. The Cayley graphs is associated to PSL 2 (Z/qZ) or PGL 2 (Z/qZ) depending on p being a quadratic residue or non-quadratic residue modulo q, whereq is a prime and q 1 mod. These are LPS Ramanujan graphs that are defined in section 3 of [6]. In what follows, we give an explicit description of the LPS Ramanujan graphs in terms of the Cayley graphs of PSL 2 (Z/mZ). Let p be a prime number such that p 1 mod and p is quadratic residue modulo m. We denote the representative of the square root of 1modmby i. We are looking at the integral solutions =(x 0,x 1,x 2,x 3 ) of the following diophantine equation (1.) x x x x 2 3 = p, where x 0 > 0 and is odd and x 1,x 2,x 3 are even numbers. There are exactly p + 1 integral solutions with such properties. To each such integral solution

4 NASER T. SARDARI, we associate the following matrix in PSL 2 (Z/mZ): apple x0 + ix (1.5) := 1 x 2 + ix 3. x 2 + ix 3 x 0 ix 1 apple x0 +ix If p is non-quadratic residue mod m, then := 1 x 2 +ix 3 /2 x 2 +ix 3 x 0 ix 1 PSL 2 (Z/mZ) and that s why the Cayley graph in this case is defined over PGL 2 (Z/mZ) and the associated Cayley graph is a bipartite graph. This gives us p+1 matrices in PSL 2 (Z/mZ). Lubotzky [7, Theorem 7..3] showed that they generate PSL 2 (Z/mZ) and the associated Cayley graph is a nonbipartite Ramanujan graph. The construction for the bipartite LPS Ramanujan graphs X p,q is similar. The only di erence is that p is non-quadratic residue modq, whereq is a prime power. Furthermore, Lubotzky showed that diamx p,m apple2log p (n)+2log p 2+1. girthx p,m 2 3 log p(n) 2log p 2. In the following theorem that is essentially due to Lubotzky, Phillips and Sarnak [6], we give a correspondence between non-backtracking path of length h from the identity vertex to vertex v h of LPS Ramanujan graph X p,m and the primitive elements of integral quaternion Hamiltonian (the gcd of the coordinates is one) of square norm p h up to units of H(Z). Theorem 1.1 (Due to Lubotzky, Phillips and Sarnak[6]). Let X p,m be LPS Ramanujan graph associated to prime number p and odd number m. Then there is a correspondence between non-backtracking paths (v 0,...,v h ) of length h from v 0 = id to v h in X p,m and the set of primitive integral solutions to the following diophantine equation (1.6) x x x x 2 = p h, x 1 v h,1 mod 2m,. x v h, mod 2m, where v h =(v h,1,...,v h, ) and the above congruence conditions hold for some modulo 2m, wheregcd(2m, )=1. In particular, finding the shortest path between v 0 and v h is reduced to finding the smallest exponent h such that diophantine equation (1.6) has an integral solution (x 1,x 2,x 3,x ). Note that solution (x 1,x 2,x 3,x ) to the minimal exponent h is necessarily primitive, otherwise by dividing by p 2 we find a smaller path of length (h 2).

5 RAMANUJAN GRAPHS AND RANDOM CAYLEY GRAPHS 5 In the bipartite case where p is non-quadratic residue modulo prime number p 0 1 mod, we show that the diameter is greater than log 3 p ( X p,p 0 )+O(1). Moreappleprecisely, we show that the distance between the identity matrix I and 0 1 W := 10 in the bipartite Ramanujan graph X p,p 0 =PGL 2(Z/p 0 Z)where p is a non-quadratic residue mod p 0 is bigger than 8 log 3 p ( X p,p 0 ) log 3 p 2, where X p,p 0 =p 03 p 0 is the number of vertices of X p,p 0. In the non-bipartite case, our theorem is weaker. We take a composite number m such that p is a quadratic residue modulo m. The vertices of the graph X p,m are associated to PSL 2 (Z/mZ). We show that the diameter of the LPS Ramanujan graphs X p,m is greater than m log 3 p (n) log p + O(1), q where n is the number of vertices of X p,m, q m, q is a prime power and q 6=m. We note that Y n = m 3 /2 (1 1/r 2 ), r m,r prime and in the proof of this result, we only use the fact that n apple m 3 /2. Similarly, for non-bipartite graphs X p,m apple, we show that either the distance apple 1 q 0 1 between the identity matrix I and I 0 := 01 or between I and W := 10 in X p,m is larger than m log 3 p ( X p,m ) log p log q 3 p 2. As a result, (1.7) log 3 p ( X p,m ) log p m q log 3 p 2 apple diam(x p,m ).

6 6 NASER T. SARDARI Theorem 1.2. Let p, p 0 be primes where p 1 mod is a non-quadratic residue modulo p 0. Moreover, assume that m and q are integers where q is a prime power that divides m and p is quadratic residue modulo m. Let X p,p 0 and X p,m be the associated bipartite and non-bipartite Ramanujan graphs. Then the diameter of the bipartite LPS Ramanujan graph X p,p 0 is larger than 8 log 3 p ( X p,p 0 ) log 3 p 2. In the non-bipartite case, the diameter of the LPS graph X p,m is larger than m (1.8) log 3 p ( X p,m ) log p log q 3 p 2 apple diam(x p,m ). The following corollary is an immediate consequence of Theorem 1.2. Corollary 1.3. Let p and q be prime numbers that are congruent to 1 mod and p>1250. Then the diameter of the LPS Ramanujan graph X p,5qk for any integer k is greater than or equal to (1.9) 3 log p X p,5qk Remark 1.. We conjecture that the diameter of LPS Ramanujan graph X p,q where q is a prime number is asymptotic to (/3)log p X p,q.weexpect that a variate of our argument gives a sharp lower bound for the diameter of X p,q by choosing vertices with large distance from the identity (e.g. W and I 0 in our argument). We give our numerical results for the distance of W from the identity vertex in Table 3. Our data comes from our algorithm that we developed and implemented for navigation on LPS Ramanujan graphs [9]. We refer the reader to [9, Remark 1.10] for further discussion of the distribution of the distance of diagonal elements from the identity where the possible vertices with large distances from the identity matrix are listed. On the other hand, we use the Ramanujan bound on the nontrivial eigenvalues of the adjacency matrix to prove the distance of almost all pairs of vertices is less than (1+ )log k (n). The archimedean version of this problem has been discussed in Sarnak s letter to Scott Aaronson and Andy Pollington [11, Page 28]. More precisely, we prove the following stronger result in Section 3: Theorem 1.5. Let G be a k-regular Ramanujan graph and fix a vertex x2v (G). Let R be an integer such that R>(1+ )log k 1 (n). DefineM(x,R) to be the set of all vertices y 2G such that there is no path from x to y with length R (we allow backtracking paths). Then,. (1.10) M(x, R) applen 1 (1 + R) 2.

7 RAMANUJAN GRAPHS AND RANDOM CAYLEY GRAPHS Outline of the paper In Section 2, we prove Theorem 1.2. The proof uses some elementary facts in Number Theory. In Section 3, we prove Theorem 1.5. As a corollary, we prove that the distance of almost all pairs of vertices is less than (1 + ) log k 1 (n). The proof is based on giving a sharp upper bound on the variance of the distance of vertices from a fixed vertex. We use the Ramanujan bound on the eigenvalues of the adjacency matrix of the graph to give an upper bound on the variance of the distance. Finally, in Section, we compute the diameter of two families of 6-regular graphs. From our numerical experiments, we expect that the diameter of the LPS Ramanujan graphs [6] is asymptotic to (1.11) log 3 p (n). We define a random 6 regular Cayley graph Z q, by considering the Cayley graph of PSL 2 (Z/qZ) relative to the generating set S = {s ± 1,s± 2,s± 3 }, where s 1,s 2,s 3 are random elements of PSL 2 (Z/qZ). From the numerical experiments, we show that in fact the random Cayley graph has a shorter diameter and break the /3log 5 n lower bound for the LPS Ramanujan graphs. For example, we obtained a sample from the random Cayley graph of PSL 2 (Z/229Z), such that (1.12) diam(z 229 ) < 1.23 log 5 n. We expect that the diameter of the random Cayley graph would be as small as possible. More precisely, for >0 (1.13) diam(z q ) apple (1 + ) log 5 (n), almost surely as q!1. Acknowledgments. I would like to thank my Ph. D. advisor, Peter Sarnak for suggesting this project to me and also his comments on the earlier versions of this work. I am also very grateful for several insightful and inspiring conversations with him during the course of this work. In addition, I would like to thank Ori Parzanchevski. Finally, I would like to thank the careful reading and comments of the anonymous referees.

8 8 NASER T. SARDARI 2. Lower bound for the diameter of the Ramanujan graphs In this section, we give the proofs of Theorem 1.1 and Theorem 1.2. Proof of Theorem (1.1). Let H(Z) denote the integral Hamiltonian quaternions H(Z) :={x 0 + x 1 i + x 2 j + x 3 k x t 2 Z, 0 apple t apple 3},i 2 = j 2 = k 2 = 1, where ij = ji=k etc. Let :=x 0 +x 1 i+x 2 j +x 3 k 2H(Z). Denote :=x 0 x 1 i x 2 j x 3 k and Norm( ):= =x 2 0 +x2 1 +x2 2 +x2 3. Let p be a prime number such that p 1 mod. We consider the set of :=x 0 +x 1 i+x 2 j+x 3 k 2H(Z) such that (2.1) Norm( ) :=x x x x 2 3 = p, and x 0 >0 is odd and x 1,x 2,x 3 are even numbers. There are exactly (p+1)/2 pairs, 2 H(Z) with such properties. We call this set LPS generator set associated to prime p and denote this set by Let S p := { 1, 1,..., (p+1)/2, (p+1)/2 }. 0 p := { 2 H: Norm( )=p k for some k 0 and 1mod2}. 0 p is closed under multiplication and if we identify 1 and 2 in 0 p whenever ±p t 1 1 =p t 2 2, t 1,t 2 2Z, then the classes so obtained form a group with [ 1][ 2] =[ 1 2] and [ ][ ] = [1]. By [6, Corollary 3.2], this group which we denote by p is free on [ 1 ],...,[ (p+1)/2 ]. The Cayley graph of p with respect to LPS generator set S p is therefore an infinite p+1-regular tree. LPS Ramanujan graphs are associated to the quotient of this infinite p + 1-regular tree by appropriate arithmetic subgroups that we describe in what follows. Let p (m) :={[ ] 2 p : = x 0 + x 1 i + x 2 j + x 3 k x 0 mod 2m}. p (m) is a normal subgroup of p.by[6, Proposition 3.3], if quadratic residues mod m, then 1 and p are p / p (m) =PSL 2 (Z/mZ).

9 RAMANUJAN GRAPHS AND RANDOM CAYLEY GRAPHS 9 The above isomorphism is defined by sending [ ] 2 p, to the following matrix in PSL 2 (Z/mZ): apple x0 + ix (2.2) := 1 x 2 + ix 3, x 2 + ix 3 x 0 ix 1 where i is a representative of square roots of 1modm. Thisidentifiesthe finite p + 1-regular graph p / p (m) by the Cayley graph of PSL 2 (Z/mZ) with respect to S p (the image of S p under the above map) that are LPS Ramanujan graph X p,m. Let (v 0,...,v h ) be a non-backtracking path of length h from v 0 =id to v h on X p,m that means that for each 1appleiappleh s i v i 1 = v i, where s i 2 S p and s i 6= s i+1 1. We lift this path to a path on the infinite p+1-regular tree p, namely: (1,s 1,s 2 s 1,...,s h...s 1 ). Let s h...s 1 =a+bi+cj +dk 2H(Z), then (2.3) a 2 + b 2 + c 2 + d 2 = p h. and (2.) a + bi + cj + dk v h mod m. Therefore, finding a shortest path from v 0 to v h is equivalent to finding a lifting of v h on the infinite tree p that is closest to the root among all other lifts. In other words, we want to find the smallest h such that there exists an integral solution (a,b,c,d) to equation (2.3) subjected to the congruence condition (2.) mod m. This completes the proof of Theorem (1.1). In the rest of this section, we give a proof of Theorem 1.2. Proof. We begin by proving the first part of the theorem. apple We show that 0 1 the distance between the identity matrix I and W := in the bipartite 10 Ramanujan graph X p,p 0 =PGL 2 (Z/p 0 Z), where p is a non-quadratic residue mod p 0 is bigger than 8 log 3 p ( X p,p 0 ) log 3 p 2.

10 10 NASER T. SARDARI By using X p,p 0 = p 03 that is smaller than p 0 the above expression simplifies to 3 log p log p p 0. p 03 p 0 We proceed by assuming the contradiction that dist(i,i 0 ) < log p p 0.There is a correspondence between non-backtracking path of length k from the identity vertex to another vertex v k of LPS Ramanujan graph X p,p 0 and the primitive elements of integral quaternion Hamiltonian (the gcd of the coordinates is one) of square norm p k up to units of H(Z); see Theorem 1.1. As aresult,dist(i,i 0 p )<log 0 p gives us a solution to the following diophantine equation (2.5) a 2 + b 2 + c 2 + d 2 = p k, where k =dist(i,i 0 ), b c d 0mod2p 0 and a 1 mod 2. At least one of b,c,d is nonzero. From this we deduce that (2.6) a 2 p k mod p 02 and p 02 apple p k. Clarify, please: That is, k =2t.? Since p is non-quadratic residue mod p 0 the above congruence identity holds only for even k. If k is even and k =2t. From 2.6 we deduce that (2.7) a ±p t mod p 02. If p t p02 2, (2.8) dist(i,i 0 )=2t log p p 0, a contradiction. Consequently, p t < p02 2.Sincea6=±pt, we deduce that (2.9) a = ±p t + lp 02 for l 6= 0. Therefore (2.10) a Hence, (2.11) 1 2 p02. p 2t p 0, and so dist(i,i 0 p 0 )=2t log p,

11 RAMANUJAN GRAPHS AND RANDOM CAYLEY GRAPHS 11 a contradiction. Hence, we conclude the first part of our theorem. Next apple we give the proof apple of the second part of our theorem. Recall that q W := 10 and I0 := and the number of vertices is 01 Y n = m 3 /2 (1 1/r 2 ) apple m 3 /2. The expression 3 log p(n) log m p q than We show that r m,r prime log p q. (2.12) max(dist(i,i 0 ), dist(i,w)) log p q. Assume the contrary that (2.13) max(dist(i,i 0 ), dist(i,w)) < log p q. 3 log p 2 simplifies and it is smaller Similarly, by using the correspondence between non-backtracking path of length k and the solutions to the associated diophantine equation for sums of four squares, dist(i,i 0 q ) < log p gives us a solution to the following diophantine equation (2.1) a 2 + b 2 + c 2 + d 2 = p k, where k =dist(i,i 0 ), b c d 0mod2q and a 1 mod 2. At least one of b,c,d is nonzero. From this we deduce that (2.15) a 2 p k mod q 2 and q 2 apple p k. We consider two cases: k even and k odd. If k is even and k =2t. Sinceq is a prime power, we deduce from 2.15 that (2.16) a ±p t mod q 2. If p t q2 2, (2.17) dist(i,i 0 )=2t log p q,

12 12 NASER T. SARDARI a contradiction. Consequently, p t < q2 2.Sincea6=±pt, we deduce that (2.18) a = ±p t + lq 2 for l 6= 0. Therefore (2.19) a Hence, (2.20) 1 2 q2. p 2t q, and so dist(i,i 0 q )=2t log p, a contradiction. Hence k is odd and k =2t+1. We want to use a similar argument to show that dist(i,w)=2t 0 +1 is an odd number. dist(i,w)</3log p (n) gives us a solution to the following diophantine equation (2.21) a 2 + b 2 + c 2 + d 2 = p k, where b a d 0modq and c 0 mod 2. Since a is odd, then q apple a. We deduce that (2.22) c 2 p k mod q 2 and q 2 apple p k. We consider two cases: k even and k odd. If k is even and k =2t. Sinceq is a prime power from 2.22 we deduce that (2.23) c ±p t mod q 2. If p t q2 2, (2.2) dist(i,w)=2t log p q, a contradiction. Consequently, p t < q2 2.Sincec is even, then c 6= ±pt.we deduce that (2.25) c = ±p t + lq 2 for l 6= 0. Therefore, (2.26) c 1 2 q2.

13 RAMANUJAN GRAPHS AND RANDOM CAYLEY GRAPHS 13 Hence, (2.27) p 2t 1 q, dist(i,w)=2t log p q. This is a contradiction. Therefore k =2t 0 +1 for some t 0. We now investigate the case where and dist(i,i 0 )=2t +1< log p q dist(i,w)=2t 0 +1< log p q. dist(i,i 0 )=2t+1 gives us a solution to the following diophantine equation (2.28) a 2 + b 2 + c 2 + d 2 = p 2t+1 < q. Where b c d 0 mod2q and a 1 mod 2. At least one of b,c,d is nonzero. Hence (2.29) q 2 <p 2t+1 <q. q dist(i,w)=2t 0 +1<log p, gives us a solution to the following diophantine equation (2.30) a b c d 2 0 = p 2t 0+1 <q /, where b 0 a 0 d 0 0modq and a 0 1 mod 2. From 2.28 and 2.30 we deduce that (2.31) a 2 p 2t+1 mod q 2 and a is odd a<p t+1/2 <q 2 /2, c 2 p 2t 0+1 mod q 2 and c is even c<p t 0+1/2 <q 2 /2. If t 0 >t, then from 2.31 we deduce that (2.32) ± ap t 0 t = c. However, this is incompatible with the parities of a and c. The case t 0 applet is treated similarly. Hence, we conclude Theorem 1.2.

14 1 NASER T. SARDARI 3. Visiting almost all points after (1 + ) log k 1 (n) steps In this section, we show that if we pick two random points from a k-regular Ramanujan graph G, almost surely they have a distance less than (3.1) (1 + ) log k 1 (n). The idea is to use the spectral gap of the adjacency matrix of the Ramanujan graphs to prove an upper bound on the variance. A similar strategy has been implemented by Sarnak; see [11, Page 28]. Proof of Theorem 1.5. Let A(x,y) be the adjacency matrix of the Ramanujan graph G, i.e. ( 1ifx y (3.2) A(x, y) :=. 0 otherwise Since A(x,y) is a symmetric matrix, it is diagonalizable. We can write the spectral expansion of this matrix by the set of its eigenfunctions. Namely, (3.3) A(x, y) = k G + X j j j(x) j (y), where { j } is the orthonormal basis of the nontrivial eigenfunctions with eigenvalues { j } for the adjacency matrix A(x,y). Since we assumed that G is a Ramanujan graph, then j apple2 p k 1. We change the variables and write (3.) j =2 p k 1 cos j, where j is a real number. We define S(R):=(k 1) R 2 U A R 2 p,where k 1 U R (x) is the Chebyshev polynomial of the second kind, i.e. (3.5) U R (x) := sin((r + 1) arccos x). sin(arccos x) The following is the spectral expansion of S(R): (3.6) (k 1) R 2 U k R 2 p k 1 S(R)(x, y) := + X (k 1) R j 2 UR G 2 p k 1 j j(x) j (y).

15 RAMANUJAN GRAPHS AND RANDOM CAYLEY GRAPHS 15 Remark 3.1. Note that if we lift the linear operator S(R) to the universal covering space of the k-regular graph G, (which is an infinite k-regular tree), then S(R) is the linear operator, which takes the sum of a function on the spheres with radius R 2k for 0applek applebr/2c; see[, Chapter 1]. Namely, X X (3.7) S(R)f(x) := f(y). 0applekapplebR/2c dist(x,y)=r 2k There is a path of length R (we allow backtracking paths) from x to y if dist(x,y)=r 2k. See[6, Remark 2] for more discussion of this operator. From the formula for the kernel of S(R) given in 3.6, we obtain (3.8) k(k 1)R 1 S(R)(x, y) = + X G j (k 1) R 2 sin((r + 1) j ) sin j j(x) j (y). We calculate the variance over y. For i 6= j, wehave P y2g i(y) j (y)=0 and P y2g i(y) 2 = 1. So only the diagonal terms remain in the following summation: (3.9) Var(x) := X y2g = X j S(R)(x, y) k(k 1) R 1 G (k 1) R (sin(r + 1) j) 2 (sin j ) 2 j(x) 2. Since { j } is an orthonormal basis, we have (3.10) 1 = 1 G + X j j(x) 2, for every x 2 G. We also have the following trivial trigonometric inequality, which is derived from the geometric series summation formula: 2 (3.11) sin(r + 1) sin = RX e i apple R +1. j=0 From 3.10 and 3.11, we obtain (3.12) Var(x) apple (R + 1) 2 (k 1) R. We define (3.13) M := {y : S(R)(x, y) =0}.

16 16 NASER T. SARDARI Note that M is the set of all vertices y 2G, such that there is no path from x to y with length R. Therefore, this is exactly the set M(x,R) as defined in the Theorem 1.5. By the definition of the Var given in 3.9, (3.1) M From 3.1 and 3.12, wehave k(k 1) R 1 G 2 apple Var(x). (3.15) M (k 1) R < G 2 (R + 1) 2. If we choose R>(1+ )log k 1 (n), then (3.16) M applen 1 (1 + R) 2. Therefore, we conclude the Theorem Numerical Results In this section, we present our numerical experiments for the diameter of the family of 6-regular LPS Ramanujan graphs X 5,q and compare it with the diameter of a family of 6-regular random Cayley graphs Z q.ournumerical experiments show that the ratio of the diameter by the logarithm diam of the number of vertices log 5 X 5,q converges to /3 as q!1 for the LPS diam log 5 Z q Ramanujan graphs X 5,q. On the other hand converges to 1 as q!1 for the random Cayley graphs Z q. We give the detailed construction of the LPS Ramanujan graphs X 5,29 in what follows. The construction of LPS Ramanujan graphs X 5,q requires that 5 and 1 to be quadratic residues mod q. From the reciprocity law we deduce that all the prime factors of q are congruent to 1 or 9 mod 20. The least q with such properties is 29. We take the integral solutions =(x 0,x 1,x 2,x 3 ) of the following diophantine equation (.1) x x x x 2 3 =5, where x 0 > 0 is odd and x 1,x 2,x 3 are even numbers. There are exactly 6 integral solutions with such properties which are listed below: {(1, ±2, 0, 0), (1, 0, ±2, 0), (1, 0, 0, ±2)}. To each such integral solution =(x 0,x 1,x 2,x 3 ), we associate the following matrix in PSL 2 Z 29Z : (.2) apple x0 + ix 1 x 2 + ix 3 x 2 + ix 3 x 0 ix 1,

17 RAMANUJAN GRAPHS AND RANDOM CAYLEY GRAPHS 17 where i is the square root of Z 6 matrices in PSL 2 S := 29Z 1 mod 29 respectively. We obtain the following apple apple apple apple apple apple , 0 10, 13 8, 13 21, 11 21, 11 8 Z 29Z which generate PSL 2. The LPS Ramanujan graph X5,29 is the Cayley Z graph of PSL 2 29Z with the generator set S. The Ramanujan graph X5,29 has vertices with diameter 8. We note that d/3 log 5 (12180)e =8. apple 10 We show the level structure of X 5,29 with root in table 1 that is the 01 same at every other root since it is a Cayley graph. We note that the girth of this graph is 9 girth(x 5,29 )=9, and this means a ball of radius in the graph X 5,29 is a tree as illustrated in Figure 1. For the family of LPS bipartite Ramanujan graphs, Biggs and Boshier determined the asymptotic behavior of the girth of these graphs; see [1]. They showed that the girth is asymptotic to 3 log k 1 (n)., r N(r) (Number of vertices of X 5,29 with distance r from apple ) Table 1. Level structure of the LPS Ramanujan graphs X 5,29 We give our numerical results for the diameter of the LPS Ramanujan graphs X 5,q for 1 apple q apple 229 in Table 2. We note that diam log 5 n are close to /3. The range for our numerical experiment with the diameter of X 5,q is small

18 18 NASER T. SARDARI Figure 1. A ball of radius in the LPS Ramanujan graphs X 5,29 since the algorithm terminates in O(q 3 ) operations. In our very recent work [9], we developed and implemented a polynomial time algorithm in log(q) that finds the shortest possible path between diagonal vertices of Ramanujan graphs X p,q under a polynomial time algorithm for factoring and a Cramer type conjecture. An important feature of our algorithm is that it has been implemented and it runs and terminates quickly; see [9, Section 6]. We give strong numerical evidence that the distance of W from I is asymptotic to /3log 5 ( X 5,q ) in Table 3. These numerical experiments are consistent with our conjectures on optimal strong approximation for quadratic forms in variables [8]. The conjecture implies that for the LPS Ramanujan graphs X p,q where p is a fixed prime number, the ratio diam(xp,q) log p 1 X p,q converges to /3 as q!1. Finally, we give our numerical experiments for the diameter of the 6-regular random Cayley graphs PSL 2 (Z/qZ). To compare the diameter of the random Cayley graphs with that of the LPS Ramanujan graphs given above, we choose the same set of integers q. We generate 8 random samples for each q, and we give the averaged ratio diam log 5 n in the last column of Table. ( means that 6 of our random samples are 8 and 2 of them are 9). We note that the empirical mean of the ratio diam(zq) log 5 Z q is decreasing in q and one can easily show that diam(z q ) 1. log 5 Z q

19 RAMANUJAN GRAPHS AND RANDOM CAYLEY GRAPHS 19 q number of vertices of X 5,q Diameter diam log 5 n Table 2. LPS Ramanujan graphs X 5,q q d:= Distance between W and I d log 5 n Table 3. LPS Ramanujan graphs X 5,q q number of vertices of Z q Diameter diam log 5 n Table. Random Cayley graphs PSL 2 Z qz with 6 generators Based on our numerical experiments, we expect that diam(zq) log 5 Z q 1 in probability as q!1 for random Cayley graphs Z q. converges to

20 20 N. T. SARDARI: RAMANUJAN GRAPHS AND RANDOM CAYLEY GRAPHS References [1] N. L. Biggs and A. G. Boshier, Note on the girth of Ramanujan graphs, J. Combin. Theory Ser. B 9 (1990), [2] B. Bollobás and W. Fernandez de la Vega: The diameter of random regular graphs, Combinatorica 2 (1982), [3] P. Chiu: Cubic ramanujan graphs, Combinatorica 12 (1992), [] G. Davidoff, P. Sarnak and A. Valette: Elementary number theory, group theory, and Ramanujan graphs, volume 55 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, [5] E. Lubetzky and Y. Peres: Cuto on all Ramanujan graphs, ArXiv e-prints, July [6] A. Lubotzky, R. Phillips and P. Sarnak: Ramanujan graphs, Combinatorica 8 (1988), [7] A. Lubotzky: Discrete groups, expanding graphs and invariant measures, Modern Birkhäuser Classics. Birkhäuser Verlag, Basel, 2010, with an appendix by Jonathan D. Rogawski, Reprint of the 199 edition. [8] N. T Sardari: Optimal strong approximation for quadratic forms, ArXiv e-prints, October [9] N. T Sardari: Complexity of strong approximation on the sphere, ArXiv e-prints, March [10] P. Sarnak: Some applications of modular forms, volume99ofcambridge Tracts in Mathematics, Cambridge University Press, Cambridge, [11] P. Sarnak: Letter to Scott Aaronson and Andy Pollington on the Solovay-Kitaev Theorem, February 2015, Naser T. Sardari A liation (fill in, please)