Ramanujan Graphs of Every Degree Adam Marcus (Crisply, Yale) Daniel Spielman (Yale) Nikhil Srivastava (MSR India) Expander Graphs Sparse, regular well-connected graphs with many properties of random graphs. Random walks mix quickly. Every set of vertices has many neighbors. Pseudo-random generators. Error-correcting codes. Sparse approximations of complete graphs. Spectral Expanders Let G be a graph and A be its adjacency matrix a 0 1 0 0 1 b 1 0 1 0 1 e 0 1 0 1 0 c 0 0 1 0 1 1 1 0 1 0 d Eigenvalues λ1 λ2 λn “trivial” ≥ ≥ ···≥ If d-regular (every vertex has d edges), λ1 = d Spectral Expanders If bipartite (all edges between two parts/colors) eigenvalues are symmetric about 0 If d-regular and bipartite, λ = d n − “trivial” a b 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 c d 1 1 0 0 0 0 0 1 1 0 0 0 e f 1 0 1 0 0 0 Spectral Expanders G is a good spectral expander if all non-trivial eigenvalues are small [ ] -d 0 d Bipartite Complete Graph Adjacency matrix has rank 2, so all non-trivial eigenvalues are 0 a b 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 c d 1 1 1 0 0 0 1 1 1 0 0 0 e f 1 1 1 0 0 0 Spectral Expanders G is a good spectral expander if all non-trivial eigenvalues are small [ ] -d 0 d Challenge: construct infinite families of fixed degree Spectral Expanders G is a good spectral expander if all non-trivial eigenvalues are small [ ] ( 0 ) -d 2pd 1 2pd 1 d − − − Challenge: construct infinite families of fixed degree Alon-Boppana ‘86: Cannot beat 2pd 1 − Ramanujan Graphs: 2pd 1 − G is a Ramanujan Graph if absolute value of non-trivial eigs 2pd 1 − [ ] ( 0 ) -d 2pd 1 2pd 1 d − − − Ramanujan Graphs: 2pd 1 − G is a Ramanujan Graph if absolute value of non-trivial eigs 2pd 1 − [ ] ( 0 ) -d 2pd 1 2pd 1 d − − − Margulis, Lubotzky-Phillips-Sarnak’88: Infinite sequences of Ramanujan graphs exist for d =prime+1 Ramanujan Graphs: 2pd 1 − G is a Ramanujan Graph if absolute value of non-trivial eigs 2pd 1 − [ ] ( 0 ) -d 2pd 1 2pd 1 d − − − Friedman’08: A random d-regular graph is almost Ramanujan : 2pd 1+✏ − Ramanujan Graphs of Every Degree Theorem: there are infinite families of bipartite Ramanujan graphs of every degree. Ramanujan Graphs of Every Degree Theorem: there are infinite families of bipartite Ramanujan graphs of every degree. And, are infinite families of (c,d)-biregular Ramanujan graphs, having non-trivial eigs bounded by pd 1+pc 1 − − Bilu-Linial ‘06 Approach Find an operation that doubles the size of a graph without creating large eigenvalues. [ ( ) ] 2pd 1 0 2pd 1 -d − − − d Bilu-Linial ‘06 Approach Find an operation that doubles the size of a graph without creating large eigenvalues. [ ( ) ] 2pd 1 0 2pd 1 -d − − − d 2-lifts of graphs a b e c d 2-lifts of graphs a a b b e e c c d d duplicate every vertex 2-lifts of graphs a0 a1 1 b0 b e0 e1 c0 c1 1 d0 d duplicate every vertex 2-lifts of graphs a0 a1 1 b0 b e0 e1 c0 c1 1 d0 d for every pair of edges: leave on either side (parallel), or make both cross 2-lifts of graphs a0 a1 1 b0 b e0 e1 c0 c1 1 d0 d for every pair of edges: leave on either side (parallel), or make both cross 2-lifts of graphs 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 0 2-lifts of graphs 0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0 2-lifts of graphs 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 Eigenvalues of 2-lifts (Bilu-Linial) Given a 2-lift of G, create a signed adjacency matrix As with a -1 for crossing edges and a 1 for parallel edges 1 a0 a 0 -1 0 0 1 1 b0 b -1 0 1 0 1 1 0 1 0 -1 0 e0 e 1 0 0 -1 0 1 c0 c 1 1 0 1 0 1 d0 d Eigenvalues of 2-lifts (Bilu-Linial) Theorem: The eigenvalues of the 2-lift are the union of the eigenvalues of A (old) and the eigenvalues of As (new) 1 a0 a 0 -1 0 0 1 1 b0 b -1 0 1 0 1 1 0 1 0 -1 0 e0 e 1 0 0 -1 0 1 c0 c 1 1 0 1 0 1 d0 d Eigenvalues of 2-lifts (Bilu-Linial) Theorem: The eigenvalues of the 2-lift are the union of the eigenvalues of A (old) and the eigenvalues of As (new) Conjecture: Every d-regular graph has a 2-lift in which all the new eigenvalues have absolute value at most 2pd 1 − Eigenvalues of 2-lifts (Bilu-Linial) Conjecture: Every d-regular graph has a 2-lift in which all the new eigenvalues have absolute value at most 2pd 1 − Would give infinite families of Ramanujan Graphs: start with the complete graph, and keep lifting. Eigenvalues of 2-lifts (Bilu-Linial) Conjecture: Every d-regular graph has a 2-lift in which all the new eigenvalues have absolute value at most 2pd 1 − We prove this in the bipartite case. a 2-lift of a bipartite graph is bipartite Eigenvalues of 2-lifts (Bilu-Linial) Theorem: Every d-regular graph has a 2-lift in which all the new eigenvalues have absolute value at most 2pd 1 − Trick: eigenvalues of bipartite graphs are symmetric about 0, so only need to bound largest Eigenvalues of 2-lifts (Bilu-Linial) Theorem: Every d-regular bipartite graph has a 2-lift in which all the new eigenvalues have absolute value at most 2pd 1 − First idea: a random 2-lift m Specify a lift by s 1 2 {± } Pick s uniformly at random First idea: a random 2-lift m Specify a lift by s 1 2 {± } Pick s uniformly at random Are graphs for which this usually fails First idea: a random 2-lift m Specify a lift by s 1 2 {± } Pick s uniformly at random Are graphs for which this usually fails Bilu and Linial proved G almost Ramanujan, implies new eigenvalues usually small. Improved by Puder and Agarwal-Kolla-Madan The expected polynomial Consider E [ χAs (x)] s The expected polynomial Consider E [ χAs (x)] s p Prove max-root E [ χAs (x)] 2 d 1 s − ⇣ ⌘ Prove χ A s ( x ) is an interlacing family Conclude there is an s so that max-root (χ (x)) 2pd 1 As − The expected polynomial Theorem (Godsil-Gutman ‘81): E [ χAs (x)]=µG(x) s the matching polynomial of G The matching polynomial (Heilmann-Lieb ‘72) n 2i i µ (x)= x − ( 1) m G − i i 0 X≥ mi = the number of matchings with i edges µ (x)=x6 7x4 + 11x2 2 G − − µ (x)=x6 7x4 + 11x2 2 G − − one matching with 0 edges µ (x)=x6 7x4 + 11x2 2 G − − 7 matchings with 1 edge µ (x)=x6 7x4 + 11x2 2 G − − µ (x)=x6 7x4 + 11x2 2 G − − Proof that E [ χAs (x)]=µG(x) s Expand E [det( xI A s )] using permutations s − x ±1 0 0 ±1 ±1! ±1 x ±1 0 0 0! 0 ±1 x ±1 0 0! 0 0 ±1 x ±1 0! ±1 0 0 ±1 x ±1! ±1 0 0 0 ±1 x! Proof that E [ χAs (x)]=µG(x) s Expand E [det( xI A s )] using permutations s − same edge: x ±1 0 0 ±1 ±1! same value ±1 x ±1 0 0 0! 0 ±1 x ±1 0 0! 0 0 ±1 x ±1 0! ±1 0 0 ±1 x ±1! ±1 0 0 0 ±1 x! Proof that E [ χAs (x)]=µG(x) s Expand E [det( xI A s )] using permutations s − same edge: x ±1 0 0 ±1 ±1! same value ±1 x ±1 0 0 0! 0 ±1 x ±1 0 0! 0 0 ±1 x ±1 0! ±1 0 0 ±1 x ±1! ±1 0 0 0 ±1 x! Proof that E [ χAs (x)]=µG(x) s Expand E [det( xI A s )] using permutations s − x ±1 0 0 ±1 ±1! ±1 x ±1 0 0 0! 0 ±1 x ±1 0 0! 0 0 ±1 x ±1 0! ±1 0 0 ±1 x ±1! ±1 0 0 0 ±1 x! Get 0 if hit any 0s Proof that E [ χAs (x)]=µG(x) s Expand E [det( xI A s )] using permutations s − x ±1 0 0 ±1 ±1! ±1 x ±1 0 0 0! 0 ±1 x ±1 0 0! 0 0 ±1 x ±1 0! ±1 0 0 ±1 x ±1! ±1 0 0 0 ±1 x! Get 0 if take just one entry for any edge Proof that E [ χAs (x)]=µG(x) s Expand E [det( xI A s )] using permutations s − x ±1 0 0 ±1 ±1! ±1 x ±1 0 0 0! 0 ±1 x ±1 0 0! 0 0 ±1 x ±1 0! ±1 0 0 ±1 x ±1! ±1 0 0 0 ±1 x! Only permutations that count are involutions Proof that E [ χAs (x)]=µG(x) s Expand E [det( xI A s )] using permutations s − x ±1 0 0 ±1 ±1! ±1 x ±1 0 0 0! 0 ±1 x ±1 0 0! 0 0 ±1 x ±1 0! ±1 0 0 ±1 x ±1! ±1 0 0 0 ±1 x! Only permutations that count are involutions Proof that E [ χAs (x)]=µG(x) s Expand E [det( xI A s )] using permutations s − x ±1 0 0 ±1 ±1! ±1 x ±1 0 0 0! 0 ±1 x ±1 0 0! 0 0 ±1 x ±1 0! ±1 0 0 ±1 x ±1! ±1 0 0 0 ±1 x! Only permutations that count are involutions Correspond to matchings The matching polynomial (Heilmann-Lieb ‘72) n 2i i µ (x)= x − ( 1) m G − i i 0 X≥ Theorem (Heilmann-Lieb) all the roots are real The matching polynomial (Heilmann-Lieb ‘72) n 2i i µ (x)= x − ( 1) m G − i i 0 X≥ Theorem (Heilmann-Lieb) all the roots are real and have absolute value at most 2pd 1 − The matching polynomial (Heilmann-Lieb ‘72) n 2i i µ (x)= x − ( 1) m G − i i 0 X≥ Theorem (Heilmann-Lieb) all the roots are real and have absolute value at most 2pd 1 − p Implies max-root E [ χAs (x)] 2 d 1 s − ⇣ ⌘ Interlacing p(x)= d (x ↵ ) Polynomial i=1 − i d 1 interlaces q(x)=Q − (x βi) i=1 − Q if ↵1 β1 ↵2 ↵d 1 βd 1 ↵d ··· − − Common Interlacing p1 ( x ) and p 2 ( x ) have a common interlacing if can partition the line into intervals so that each interval contains one root from each poly β βd 1 2 − β1 β3 Common Interlacing p1 ( x ) and p 2 ( x ) have a common interlacing if can partition the line into intervals so that each interval contains one root from each poly β βd 1 ) ( ) 2( ) ( ) −( β1 β3 Common Interlacing If p1 and p2 have a common interlacing, max-root (pi) max-root (Ei [ pi ]) for some i.