Lecture 18. Ramanujan Graphs continued

Transcription

1 Stanford University Winter 218 Math 233A: Non-constructive methods in combinatorics Instructor: Jan Vondrák Lecture date: March 8, 218 Original scribe: László Miklós Lovász Lecture 18 Ramanujan Graphs continued 181 Interlacing polynomials Last time, we proved the following lemma Lemma 181 Two real-rooted, monic polynomials of the same degree, f and g, have a common interlacing if and only if for every t, 1, h t = tf + 1 t)g is real-rooted We can extend this to multiple monic polynomials as follows Suppose f 1, f 2,, f m are monic polynomials with the same degree n Let R k be the set of k-th largest roots of the polynomials counting multiple roots multiple times) We say that the polynomials have a common interlacing if there exist γ n γ n 1 γ such that for each k, R k γ k, γ k 1 Then it is not difficult to extend Lemma 181 to the following claim: Claim 182 Suppose f 1, f 2,, f m are monic polynomials of the same degree Then the following are equivalent: 1 f 1, f 2,, f m have a common interlacing 2 ach pair of polynomials f i, f j has a common interlacing 3 There exists i < j such that for the pair of polynomials f i, f j, and for any t, 1, tf i + 1 t)f j is real rooted and has a common interlacing with the rest of the polynomials 4 For any {t i } m i=1 with each t i and i t i = 1, we have i t if i is real rooted 182 Interlacing families Let S i for i from 1 to m be finite sets with a corresponding distribution µ i on them We will be working with families of polynomials f s1,s 2,,s m, indexed by s i S i We will define f s1,s 2,,s k x) = s k µ k s m µ m f s1,s 2,,s m x) We say that the family of polynomials forms an interlacing family if for any k < m, and s 1 S 1, s 2 S 2,, s k S k, the polynomials {f s1,s 2,,s k,s} s Sk have a common interlacing This allows us to prove the following theorem Theorem 183 MSS 13 If {f s1,s 2,,s m } form an interlacing family, then we can find s 1 S 1, s 2 S 2,,s m S m such that maxrootf s1,s 2,,s m ) maxrootf x))

2 Proof: We will prove that for every k m, and every s 1, s 2,, s k 1, there exists s k S k such that maxrootf s1,s 2,,s k ) f s1,s 2,,s k 1 This is true because f s1,s 2,,s k 1 is a convex combination of f s1,s 2,,s k 1,s k over various choices of s k This implies that the maximum root of f s1,s 2,,s k 1 is a convex combination of the maximum roots of f s1,s 2,,s k 1,s k over various choices of s k, in particular, its maximum root is larger than at least on one of the maximum roots of f s1,s 2,,s k Applying this to each k, the theorem is proved Let us now turn to adjacency matrices of graphs Given a graph G = V, ) and a vector s {, 1}, the graph G 2) s is the two-lift of G corresponding to the vector s, where a coordinate indicates non-crossing edges and a 1 coordinate indicates crossing edges Then if A is the adjacency matrix of G, then A s will be obtained from A by writing 1) se for each spot corresponding to a 1 entry in A recall that A has only and 1 entries) Then let us denote the polynomial χ sx) = detxi A s ) We saw last time that χ x) = χ s x) = M G x), s 1,s 2,,s m where M G x) denotes the matching polynomial of G By Heilmann-Lieb, we know that the maximum root of the matching polynomial is at most 2 d 1 We would like to conclude using Theorem 183 that this implies that there exists a choice of s such that the maximum root is 2 d 1 We thus have to prove that the polynomials form an interlacing family, where we recall that we denote First, let us prove the following: χ s1,s 2,s k x) = Claim 184 For p 1, p 2,, p m, 1, define χ p1,p 2,,p m x) = Then all such polynomials are real rooted detxi A s ) s k,,s m p i s 1,s 2,,s m {,1} i:s i =1 i:s i = 1 p i )χ s x) This claim implies that the polynomials form an interlacing family Indeed, it is easy to see that χ s1,s 2,,s k x) = χ s1,s 2,,s k, 1 2, 1 2,, 1 2 x) Thus, this implies that any convex combination of χ s1,s 2,,s k 1, and χ s1,s 2,,s k 1,1 is real rooted since it is equal to χ s1,s 2,,s k 1,t, 1 2, 1 2,, 1 2 In order to prove the claim, we prove the following theorem: Theorem 185 If v 1, v 2,, v m are random vectors in C n with finite support, then m detxi v i vi ) is a real stable polynomial v 1,v 2,,v m i=1

3 For the case of Ramanujan graphs, χ sx) = detxi A s ), where we can write A s = di e b e b e for vectors b e = if s e =, b e = if s e = 1 1 Thus, Theorem 185 implies Claim 184 We will be using a few facts from linear algebra Fact 1 If A C n n is a nonsingular matrix, and u, v C n are vectors, then A + uv ) = 1 + v A 1 u) deta) 181) Proof: First, assume A = I Then we have ) ) ) ) I u I I + uv u I 1 + v = u v 1 1 v 1 Taking the determinant of both sides, we obtain 181) In general, we have, for u = A 1 u, deta + uv ) = deta) deti + u v ) = deta)1 + v u ) = 1 + v A 1 u) deta) Recall that given a matrix A, the trace is the sum along the diagonal TrA) = i a ii Also recall that for two matrices A and B, TrAB) = i,j a ij b ji = TrBA) Fact 2 If A, B, then TrAB) Proof: We can write A = i α iu i u i for α i, and B = β j j v jv j for β j Then TrAB) = i,j α i β j Tru i u i v j v j ) = i,j α i β j Trv j u i u i v j ) = i,j α i β j v j u i 2 Fact 3 For non-singular A C n n, and arbitrary B C n n, we have d deta + tb)) = deta) TrA 1 B) dt t=

4 Proof: We may assume that A = I Let B = i,j b ije i e j Then d deti + t b ij e i e dt j) = i,j t= i,j = deti + tij b ij e i e t j) ) i,j ij t ij deti + t i j b i j e i e j ) i,j tij = = i,j 1 + t ij b ij δ i,j ) t ij tij = tij = = i b ii = TrB) Proof:of Theorem 185, simplified by James Lee We first prove the following lemma Lemma 186 If A C n n is a fixed matrix, and v C n is a random vector, then deta vv ) = 1 d ) det A + t vv ) dt v t= Proof: First, assume A is non-singular Then by 181) we have deta vv ) = deta)1 v A 1 v) = deta)1 TrA 1 vv ) Thus, By Fact 3, we have deta vv ) = deta)1 TrA 1 vv )) v d dt deta + t vv )) = deta) TrA 1 vv ), t= which proves the lemma for non-singular A For singular A, we can take a sequence A ɛ A of nonsingular matrices The equation will hold for each A ɛ, and since both sides of the equation are continuous, the lemma must also hold for A For the proof of the theorem, we need one more claim ) Claim 187 The operator 1 z i preserves stability of fz 1, z 2,, z m ) Proof: Next time We have, for independent vectors v 1, v 2,, v m, v 1,,v m deta i v i v i ) = 1 ) 1 ) deta + t 1 t m i t i v i vi ) t1 =,,t m= We thus have xi i v i v i = 1 ) 1 ) detxi + t 1 t m i t i v i vi ) t1 =,,t m=

5 We know that detxi + i t i v i v i ) is stable because each v i vi is positive semidefinite using a theorem from Lecture 13) Since the operators preserve stability, we thus have that detxi i v i v i ) is stable, and so the theorem is proved