Weighted tree enumeration of cubical complexes

Transcription

1 Ghodratollah Aalipour 1 Art Duval 2 Jeremy Martin 3 1 University of Colorado, Denver 2 University of Texas at El Paso 3 University of Kansas Joint Mathematics Meetings AMS Special Session on Algebraic and Topological Methods in Combinatorics Seattle, WA January 9, 2016

2 Definition Cubical complexes Faces of Q n, n-dimensional cube: (0, 1, )-strings of length n. Dimension is number of * s. Vertices: (0, 1)-strings of length n Edge in direction i: single * in position i. Boundary: faces with one * converted to 0 or 1. *1 0* ** 1* *0 Cubical Complex: Subset of faces of Q n such that if a face is included, then so is its boundary.

3 Definition 010 * * 111 0* * * *00 1*0 100.

4 Definition Let Q be a d-dimensional cubical complex. Υ Q is a cubical spanning tree of Q when: 0. Υ (d 1) = Q (d 1) ( spanning ); 1. Hd 1 (Υ; Z) is a finite group ( connected ); 2. H d (Υ; Z) = 0 ( acyclic ); 3. f d (Υ) = f d (Q) β d (Q) ( count ). If 0. holds, then any two of 1., 2., 3. together imply the third condition. When d = 1, coincides with usual definition. Works more generally for cellular complexes. Inspired by ideas of Kalai on complete simplicial complex.

5 Definition Let Q be a d-dimensional cubical complex. Υ Q is a cubical spanning tree of Q when: 0. Υ (d 1) = Q (d 1) ( spanning ); 1. Hd 1 (Υ; Z) is a finite group ( connected ); 2. H d (Υ; Z) = 0 ( acyclic ); 3. f d (Υ) = f d (Q) β d (Q) ( count ). If 0. holds, then any two of 1., 2., 3. together imply the third condition. When d = 1, coincides with usual definition. Works more generally for cellular complexes. Inspired by ideas of Kalai on complete simplicial complex.

6 Definition Let Q be a d-dimensional cubical complex. Υ Q is a cubical spanning tree of Q when: 0. Υ (d 1) = Q (d 1) ( spanning ); 1. Hd 1 (Υ; Z) is a finite group ( connected ); 2. H d (Υ; Z) = 0 ( acyclic ); 3. f d (Υ) = f d (Q) β d (Q) ( count ). If 0. holds, then any two of 1., 2., 3. together imply the third condition. When d = 1, coincides with usual definition. Works more generally for cellular complexes. Inspired by ideas of Kalai on complete simplicial complex.

7 Definition Let Q be a d-dimensional cubical complex. Υ Q is a cubical spanning tree of Q when: 0. Υ (d 1) = Q (d 1) ( spanning ); 1. Hd 1 (Υ; Z) is a finite group ( connected ); 2. H d (Υ; Z) = 0 ( acyclic ); 3. f d (Υ) = f d (Q) β d (Q) ( count ). If 0. holds, then any two of 1., 2., 3. together imply the third condition. When d = 1, coincides with usual definition. Works more generally for cellular complexes. Inspired by ideas of Kalai on complete simplicial complex.

8 Definition Let s count spanning trees of each of our examples: w/o red edge 3 3 w/red edge 15 total

9 Definition Let s count spanning trees of each of our examples: w/o red edge w/o red square 3 3 w/red edge 5 5 w/red square 15 total 35 total

10 Definition Let s count spanning trees of each of our examples: w/o red edge w/o red square 3 3 w/red edge 5 5 w/red square 15 total 35 total = 82944

11 Definition 0101 * y 1 z 2 y 3 z 4 q 1 z 2 y 3 z 4 z 1 z 2 y 3 z 4 0*01 **01 1* y 1 q 2 y 3 z 4 q 1 q 2 y 3 z 4 z 1 q 2 y 3 z y 1 q 2 y 3 z 1 q 2 z 3 z 1 q 2 y 3 y 1 y 2 y 3 z q 1 y 2 y 3 z 4 *001 z 1 y 2 y 3 z z 1 y 2 q q 1 y 2 y (y 1 q 2 y 3 )(z 1 q 2 z 3 )(z 1 y 2 q 3 )(q 1 y 2 y 3 )(z 1 q 2 y 3 )

12 Definition 0101 * y 1 z 2 y 3 z 4 q 1 z 2 y 3 z 4 z 1 z 2 y 3 z 4 0*01 **01 1* y 1 q 2 y 3 z 4 q 1 q 2 y 3 z 4 z 1 q 2 y 3 z y 1 q 2 y 3 z 1 q 2 z 3 z 1 q 2 y 3 y 1 y 2 y 3 z q 1 y 2 y 3 z 4 *001 z 1 y 2 y 3 z z 1 y 2 q q 1 y 2 y (y 1 q 2 y 3 )(z 1 q 2 z 3 )(z 1 y 2 q 3 )(q 1 y 2 y 3 )(z 1 q 2 y 3 )

13 Some motivation from graphs Let G be a graph with n vertices; let (G) is the oriented boundary matrix (oriented incidence matrix); let be the Laplacian of G. L = (G) T (G) Theorem (Kirchoff s Matrix-Tree) The number of spanning trees of graph G is (λ 1 λ 2 λ n 1 )/n where 0, λ 1, λ 2,..., λ n 1 are the eigenvalues of L.

14 Example *00 *10 0*0 1*0 1*1 10* 11* = * * L = *0 1*1 1* * *00 eigenvalues(l): 5,3,3,2,1,0; spanning trees: ( )/6 = 15

15 Example *00 *10 0*0 1*0 1*1 10* 11* = * * L = *0 1*1 1* * *00 eigenvalues(l): 5,3,3,2,1,0; spanning trees: ( )/6 = 15

16 Weighted Laplacian (arbitrary dimension) weighted boundary map weighted Laplacian ˆ = σ ρ ± σ/ρ σ ρ = q i x i or q i y i ˆ k = D 1 k 1 kd k, where D k = diag(..., σi k,...). (Note that this forms a chain complex: ˆ k 1 ˆ k = 0.) ˆL k 1 = ˆ k ˆ T k = D 1 k 1 kd 2 k T k D 1 k 1

17 Weighted Laplacian (arbitrary dimension) weighted boundary map weighted Laplacian ˆ = σ ρ ± σ/ρ σ ρ = q i x i or q i y i ˆ k = D 1 k 1 kd k, where D k = diag(..., σi k,...). (Note that this forms a chain complex: ˆ k 1 ˆ k = 0.) ˆL k 1 = ˆ k ˆ T k = D 1 k 1 kd 2 k T k D 1 k 1

18 Weighted Laplacian (arbitrary dimension) weighted boundary map weighted Laplacian ˆ = σ ρ ± σ/ρ σ ρ = q i x i or q i y i ˆ k = D 1 k 1 kd k, where D k = diag(..., σi k,...). (Note that this forms a chain complex: ˆ k 1 ˆ k = 0.) ˆL k 1 = ˆ k ˆ T k = D 1 k 1 kd 2 k T k D 1 k 1

19 Weighted Matrix-Tree Theorem Recall ˆL k 1 = ˆ k 1 ˆ T k 1 Let ˆπ k := product of nonzero eigenvalues of ˆL k 1 [pdet] Let X (k 1) := product of all faces of dimension k 1 Enumerate spanning trees by ˆτ k := Υ T (Q) H k 1 (Υ) 2 wt(υ) Theorem (ADM, slight genzn of MMRW) ˆπ k = ˆτ k ˆτ k 1 H k 1 (Q) X (k 1)

20 Weighted Matrix-Tree Theorem Recall ˆL k 1 = ˆ k 1 ˆ T k 1 Let ˆπ k := product of nonzero eigenvalues of ˆL k 1 [pdet] Let X (k 1) := product of all faces of dimension k 1 Enumerate spanning trees by ˆτ k := Υ T (Q) H k 1 (Υ) 2 wt(υ) Theorem (ADM, slight genzn of MMRW) ˆπ k = ˆτ k ˆτ k 1 H k 1 (Q) X (k 1)

21 Weighted Matrix-Tree Theorem Recall ˆL k 1 = ˆ k 1 ˆ T k 1 Let ˆπ k := product of nonzero eigenvalues of ˆL k 1 [pdet] Let X (k 1) := product of all faces of dimension k 1 Enumerate spanning trees by ˆτ k := Υ T (Q) H k 1 (Υ) 2 wt(υ) Theorem (ADM, slight genzn of MMRW) ˆπ k = ˆτ k ˆτ k 1 H k 1 (Q) X (k 1)

22 Weighted Matrix-Tree Theorem Recall ˆL k 1 = ˆ k 1 ˆ T k 1 Let ˆπ k := product of nonzero eigenvalues of ˆL k 1 [pdet] Let X (k 1) := product of all faces of dimension k 1 Enumerate spanning trees by ˆτ k := Υ T (Q) H k 1 (Υ) 2 wt(υ) Theorem (ADM, slight genzn of MMRW) ˆπ k = ˆτ k ˆτ k 1 H k 1 (Q) X (k 1)

23 Weighted enumeration of spanning trees of Q n,k Conjecture (D-Klivans-M) n 1 ˆτ k (Q n ) = q [n] i=k 1 ( n 1 i where u S = i S q i( 1 y i + 1 z i ) )( k 2) i 1 S [n] S >k (u S y S z S ) ( S 2 k 1 ) Proof. Relies on linear algebra tricks from ˆ making chain complex, and eigenvalues of Q n (DKM). Example ˆτ 2 (Q 4 ) = q 7 [4] (u 123y 123 z 123 ) 1 (u 234 y 234 z 234 ) 1 (u [4] y [4] z [4] ) 2

24 Weighted enumeration of spanning trees of Q n,k Theorem (ADM) n 1 ˆτ k (Q n ) = q [n] i=k 1 ( n 1 i where u S = i S q i( 1 y i + 1 z i ) )( k 2) i 1 S [n] S >k (u S y S z S ) ( S 2 k 1 ) Proof. Relies on linear algebra tricks from ˆ making chain complex, and eigenvalues of Q n (DKM). Example ˆτ 2 (Q 4 ) = q 7 [4] (u 123y 123 z 123 ) 1 (u 234 y 234 z 234 ) 1 (u [4] y [4] z [4] ) 2

25 Weighted enumeration of spanning trees of Q n,k Theorem (ADM) n 1 ˆτ k (Q n ) = q [n] i=k 1 ( n 1 i where u S = i S q i( 1 y i + 1 z i ) )( k 2) i 1 S [n] S >k (u S y S z S ) ( S 2 k 1 ) Proof. Relies on linear algebra tricks from ˆ making chain complex, and eigenvalues of Q n (DKM). Example ˆτ 2 (Q 4 ) = q 7 [4] (u 123y 123 z 123 ) 1 (u 234 y 234 z 234 ) 1 (u [4] y [4] z [4] ) 2