An algebraic analogue of a formula of Knuth Lionel Levine (MIT) - PowerPoint PPT Presentation

An algebraic analogue of a formula of Knuth Lionel Levine (MIT) FPSAC, San Francisco, August 6, 2010 Lionel Levine an algebraic analogue of a formula of Knuth

Talk Outline ◮ Knuth’s formula: generalizing n n − 1 . ◮ ... with weights: generalizing ( x 1 + ... + x n ) n − 1 . ◮ ... with group structure: generalizing ( Z / n Z ) n − 1 . ◮ Recent developments! Lionel Levine an algebraic analogue of a formula of Knuth

Starting Point: Cayley’s Theorem ◮ The number of rooted trees on n labeled vertices is n n − 1 . ◮ Refinement: The number of trees with degree sequence ( d 1 ,..., d n ) is the coefficient of x d 1 1 ... x d n in n nx 1 ... x n ( x 1 + ... + x n ) n − 2 . ◮ We can break this out by root: n x r ( x 1 + ... + x n ) n − 2 r =1 ∏ ∑ · x i i � = r outdegrees indegrees Lionel Levine an algebraic analogue of a formula of Knuth

Oriented Spanning Trees An oriented spanning tree of K 3 , 3 . ◮ Let G = ( V , E ) be a finite directed graph. ◮ An oriented spanning tree of G is a subgraph T = ( V , E ′ ) such that ◮ one vertex, the root, has outdegree 0; ◮ all other vertices have outdegree 1; ◮ T has no oriented cycles v 1 → v 2 → ··· → v k → v 1 . Lionel Levine an algebraic analogue of a formula of Knuth

Complexity of A Directed Graph ◮ The number κ ( G ) = # of oriented spanning trees of G is sometimes called the complexity of G . ◮ Examples: κ ( K n ) = n n − 1 κ ( K m , n ) = ( m + n ) m n − 1 n m − 1 κ ( DB n ) = 2 2 n − 1 Lionel Levine an algebraic analogue of a formula of Knuth

The De Bruijn Graph DB n 00 0 01 10 1 11 DB 0 DB 1 DB 2 ◮ vertices { 0 , 1 } n , edges { 0 , 1 } n +1 . ◮ The endpoints of the edge e = b 1 ... b n +1 are its prefix and suffix: e b 1 ... b n − → b 2 ... b n +1 . Lionel Levine an algebraic analogue of a formula of Knuth

Directed Line Graphs ◮ G = ( V , E ) : finite directed graph ◮ s , t : E → V e ◮ Edge e is directed like this: s ( e ) − → t ( e ) ◮ The directed line graph L G = ( E , E 2 ) of G has ◮ Vertex set E , the edge set of G . ◮ Edge set E 2 = { ( e , f ) ∈ E × E | s ( f ) = t ( e ) } . e f e f e • − →• − → • • ← −• − → • • − → • f ( e , f ) ∈ E 2 ∈ E 2 • − → • ( e , f ) / ∈ E 2 ( e , f ) / Lionel Levine an algebraic analogue of a formula of Knuth

A Graph G and Its Directed Line Graph L G c b a a b c Lionel Levine an algebraic analogue of a formula of Knuth

Examples of Directed Line Graphs ◮ � K n = L (one vertex with n loops). ◮ � K m , n = L (two vertices { a , b } with m edges a → b and n edges b → a ). ◮ DB n = L ( DB n − 1 ). ◮ Iterated line graphs: L n G = ( E n , E n +1 ), where E n = { directed paths of n edges in G } . Lionel Levine an algebraic analogue of a formula of Knuth

Spanning Tree Enumerators ◮ Let ( x v ) v ∈ V and ( x e ) e ∈ E be indeterminates, and let κ edge ( G , x ) = ∑ T ∏ x e e ∈ T κ vertex ( G , x ) = ∑ T ∏ x t ( e ) e ∈ T The sums are over all oriented spanning trees T of G . ◮ Example: κ vertex ( K n , x ) = ( x 1 + ··· + x n ) n − 1 . Lionel Levine an algebraic analogue of a formula of Knuth

Knuth’s Formula ◮ G = ( V , E ) : finite directed graph with no sources ◮ outdegrees a 1 ,..., a n ◮ indegrees b 1 ,..., b n ≥ 1 ◮ L G : the directed line graph of G ◮ Theorem (Knuth, 1967). For any edge e ∗ of G , n a b i − 1 ∏ κ ( L G , e ∗ ) = α ( G , e ∗ ) i i =1 where α ( G , e ∗ ) = κ ( G , t ( e ∗ )) − 1 ∑ κ ( G , s ( e )) . a ∗ t ( e )= t ( e ∗ ) e � = e ∗ and a ∗ is the outdegree of t ( e ∗ ). Lionel Levine an algebraic analogue of a formula of Knuth

Weighted Knuth’s Formula ◮ G : finite directed graph with no sources ◮ L G : its directed line graph ◮ b 1 ,..., b n ≥ 1 : the indegrees of G . ◮ Theorem (L.) � b i − 1 � κ vertex ( L G , x ) = κ edge ( G , x ) ∏ ∑ x e . i ∈ V s ( e )= i ◮ Both sides are polynomials in the edge variables x e . Lionel Levine an algebraic analogue of a formula of Knuth

Specializing x e = 1 ◮ Complexity of a line graph: n a b i − 1 ∏ κ ( L G ) = κ ( G ) . i i =1 ◮ Examples: ◮ G = one vertex with n loops, L G = K n , get n n − 1 . ◮ G = two vertices, L G = K m , n , get ( m + n ) m n − 1 n m − 1 . ◮ G = DB n − 1 , L G = DB n : κ ( DB n ) = κ ( DB n − 1 ) · 2 2 n − 1 = κ ( DB n − 2 ) · 2 2 n − 1 · 2 2 n − 2 = ... = 2 2 n − 1 . Lionel Levine an algebraic analogue of a formula of Knuth

Rooted Version ◮ Fix an edge e ∗ = ( w ∗ , v ∗ ) of G . ◮ Let b ∗ be the indegree of v ∗ . ◮ Theorem (L.) If b i ≥ 1 for all i , and b ∗ ≥ 2, then � b i − 1 � κ vertex ( L G , e ∗ , x ) = x e ∗ κ edge ( G , w ∗ , x ) ∏ i ∈ V ∑ s ( e )= i x e . ∑ s ( e )= v ∗ x e Lionel Levine an algebraic analogue of a formula of Knuth

Matrix-Tree Theorem κ edge ( G , x ) = [ t ]det( t · Id − ∆ edge ) . κ vertex ( G , x ) = [ t ]det( t · Id − ∆ vertex ) . ◮ Goal: relate ∆ edge with ∆ vertex . G L G Lionel Levine an algebraic analogue of a formula of Knuth

The Missing Link: Directed Incidence Matrices ◮ Consider the K -linear maps A : K E → K V , B : K V → K E v �→ ∑ e �→ t ( e ) x e e . s ( e )= v Then ∆ edge = AB − D G ∆ vertex = BA − D L L G where D and D L are the diagonal matrices � � � � D L ( e ) = ∑ ∑ D ( v ) = x e v , x f e . s ( e )= v s ( f )= t ( e ) Lionel Levine an algebraic analogue of a formula of Knuth

Intertwining ∆ edge and ∆ vertex G L G = A ( BA − D L ) = ABA − DA = ( AB − D ) A = ∆ edge A ∆ vertex A L G G ◮ In particular ∆ vertex (ker A ) ⊂ ker A . L G ◮ Writing K E = ker A ⊕ Im ( A T ) puts ∆ vertex in block triangular L G form. Lionel Levine an algebraic analogue of a formula of Knuth

The Proof Falls Into Place ◮ Since G has no sources, A : K E → K V is onto. ◮ So AA T has full rank. ◮ So A : Im ( A T ) → K V is an isomorphism. | Im ( A T ) ) = det∆ edge ◮ So det(∆ vertex = κ ( G , x ). L G G ◮ Eigenvalues of ∆ vertex | ker A are ∑ s ( e )= i x e , each with L G multiplicity b i − 1. Lionel Levine an algebraic analogue of a formula of Knuth

Comparison with Knuth ◮ Knuth’s formula involved the strange quantity α ( G , e ∗ ) = κ ( G , t ( e ∗ )) − 1 ∑ κ ( G , s ( e )) . a ∗ t ( e )= t ( e ∗ ) e � = e ∗ ◮ Why is it missing from our formulas? Lionel Levine an algebraic analogue of a formula of Knuth

The Unicycle Lemma ◮ A unicycle of G is an oriented spanning tree together with an outgoing edge from the root. ◮ By counting unicycles through v ∗ in two ways, we get: ◮ Lemma. κ edge ( G , v ∗ , x ) ∑ x e = ∑ κ edge ( G , s ( e ) , x ) x e . s ( e )= v ∗ t ( e )= v ∗ ◮ So Knuth’s formula simplifies to n κ ( L G , e ∗ ) = 1 a b i − 1 ∏ κ ( G , s ( e ∗ )) . i a ∗ i =1 Lionel Levine an algebraic analogue of a formula of Knuth

The Sandpile Group of a Graph ◮ K ( G , v ∗ ) ≃ Z n − 1 / ∆ Z n − 1 , where ∆ = D − A is the reduced Laplacian of G . ◮ Lorenzini ’89/’91 (“group of components”), Dhar ’90, Biggs ’99 (“critical group”), Baker-Norine ’07 (“Jacobian”). ◮ Directed graphs: Holroyd et al. ’08 ◮ Matrix-tree theorem : # K ( G , v ∗ ) = det ∆ = # { spanning trees of G rooted at v ∗ } . ◮ Choice of sink: K ( G , v ∗ ) ≃ K ( G , v ′ ∗ ) if G is Eulerian. Lionel Levine an algebraic analogue of a formula of Knuth

Maps Between Sandpile Groups Theorem (L.) If G is Eulerian, then the map Z E → Z V e �→ t ( e ) descends to a surjective group homomorphism K ( L G , e ∗ ) → K ( G , t ( e ∗ )) . Lionel Levine an algebraic analogue of a formula of Knuth

Maps Between Sandpile Groups Theorem (L.) If G is balanced k -regular, then the map Z V → Z E v �→ ∑ e s ( e )= v descends to an isomorphism of groups K ( G ) ≃ k K ( L G ) . ◮ Analogous to results of Berget, Manion, Maxwell, Potechin and Reiner on undirected line graphs. arXiv:0904.1246 Lionel Levine an algebraic analogue of a formula of Knuth

The Sandpile Group of DB n ◮ De Bruijn Graph DB n = L n (a single vertex with 2 loops). ◮ Theorem (L.) n − 1 ( Z / 2 j Z ) 2 n − 1 − j . � K ( DB n ) = j =1 ◮ Generalized by Bidkhori and Kishore to k -ary De Bruijn graphs for any k . Lionel Levine an algebraic analogue of a formula of Knuth

Equating Exponents ◮ By counting spanning trees, we know that # K ( DB n ) = κ ( DB n , v ∗ ) = 2 2 n − n − 1 . ◮ Now write K ( DB n ) = Z a 1 2 ⊕ Z a 2 4 ⊕ Z a 3 8 ⊕ ... ⊕ Z a m 2 m for some nonnegative integers m and a 1 ,..., a m satisfying m ja j = 2 n − n − 1 . ∑ (1) j =1 ◮ By the previous theorem and inductive hypothesis K ( DB n − 1 ) ≃ 2 K ( DB n ) Z 2 n − 3 ⊕ Z 2 n − 4 ⊕···⊕ Z 2 n − 2 ≃ Z a 2 2 ⊕ Z a 3 4 ⊕ ... ⊕ Z a m 2 m − 1 . 2 4 ◮ So m = n − 1 and a j = 2 n − j − 1 . Lionel Levine an algebraic analogue of a formula of Knuth

A (Formerly) Open Problem From EC1 ◮ In EC1, Stanley asks for a bijection { pairs of binary De Bruijn sequences of order n } � { all binary sequences of length 2 n } ◮ Both sets have cardinality 2 2 n . Lionel Levine an algebraic analogue of a formula of Knuth