Simplicial and Cellular Spanning Trees, I: General Theory

Transcription

1 Simplicial and Cellular Spanning Trees, I: General Theory Art Duval (University of Texas at El Paso) Caroline Klivans (Brown University) Jeremy Martin (University of Kansas) University of California, Davis March 20

2 Definitions Counting Spanning Trees Weighted Spanning Tree Enumerators Graphs and Spanning Trees G = (V, E): simple connected graph Definition A spanning tree of G is a subgraph (V, T ) such that 1. (V, T ) is connected (every pair of vertices is joined by a path); 2. (V, T ) is acyclic (contains no cycles); 3. T = V 1. Any two of these conditions together imply the third.

3 Definitions Counting Spanning Trees Weighted Spanning Tree Enumerators Graphs and Spanning Trees G

4 Definitions Counting Spanning Trees Weighted Spanning Tree Enumerators Graphs and Spanning Trees G T

5 Definitions Counting Spanning Trees Weighted Spanning Tree Enumerators Counting Spanning Trees Let τ(g) denote the number of spanning trees of G. Graph G τ(g) Any tree 1 C n (cycle on n vertices) n K n (complete graph on n vertices) n n 2 (Cayley) K p,q (complete bipartite graph) p q 1 q p 1 (Fiedler-Sedláçek) Q n (n-dimensional hypercube) n 2 2n k 1 k (n k) k=2

6 Definitions Counting Spanning Trees Weighted Spanning Tree Enumerators The Laplacian Matrix Let G = (V, E) be a graph with V = [n] = {1, 2,..., n}. Definition The Laplacian of G is the n n matrix L = [l ij ]: deg G (i) if i = j, l ij = 1 if i, j are adjacent, 0 otherwise. L is a real symmetric matrix L = MM tr, where M is the signed incidence matrix of G

7 Definitions Counting Spanning Trees Weighted Spanning Tree Enumerators The Matrix-Tree Theorem Matrix-Tree Theorem, Version I: Let 0, λ 1, λ 2,..., λ n 1 be the eigenvalues of L. Then τ(g) = λ 1λ 2 λ n 1. n Matrix-Tree Theorem, Version II: Let L i be the reduced Laplacian obtained by deleting the i th row and i th column of L. Then τ(g) = det L i. [Kirchhoff, 1847]

8 Definitions Counting Spanning Trees Weighted Spanning Tree Enumerators The Matrix-Tree Theorem Sketch of proof: 1. Expand det L i using the Binet-Cauchy formula: det L i = det M i Mi tr = (det M T ) 2 T E T =n 1 where M T = square submatrix of M i with columns T 2. Show that det M T = { ±1 if T is acyclic, 0 otherwise.

9 Definitions Counting Spanning Trees Weighted Spanning Tree Enumerators Weighted Spanning Tree Enumerators Idea: Let s record combinatorial information about a spanning tree T by assigning it a monomial weight x T. (e.g., vertex degrees; number of edges in specified sets; etc.) Definition The weighted spanning tree enumerator of G is the generating function T T (G) where T (G) denotes the set of spanning trees of G. x T

10 Definitions Counting Spanning Trees Weighted Spanning Tree Enumerators Weighted Spanning Tree Enumerators The weighted spanning tree enumerator of a graph reveals much more detailed combinatorial information about spanning trees of G than merely counting them (particularly when it factors!) can suggest bijective proofs of formulas for τ(g)

11 Definitions Counting Spanning Trees Weighted Spanning Tree Enumerators The Weighted Laplacian Introduce an indeterminate e ij for each pair of vertices i, j. Set e ij = e ji, and if i, j are not adjacent, then set e ij = 0. The weighted Laplacian of G is the n n matrix ˆL = [ˆl ij ], where e ij if i = j, ˆl ij = j i e ij if i j. Setting e ij = 1 for each edge ij recovers the usual Laplacian L.

12 Definitions Counting Spanning Trees Weighted Spanning Tree Enumerators The Weighted Matrix-Tree Theorem Weighted Matrix-Tree Theorem I: If 0, ˆλ 1, ˆλ 2,..., ˆλ n 1 are the eigenvalues of ˆL, then T T (G) ij T e ij = ˆλ 1ˆλ 2 ˆλ n 1. n Weighted Matrix-Tree Theorem II: If ˆL kl is obtained by deleting the k th row and l th column of ˆL, then e ij = ( 1) k+l det ˆL kl. T T (G) ij T

13 Definitions Counting Spanning Trees Weighted Spanning Tree Enumerators Example: The Cayley-Prüfer Theorem Weight spanning trees of complete graph K n by degree sequence: x T = n i=1 x deg T (i) i Theorem [Cayley-Prüfer] x T = (x 1 x 2 x n )(x 1 + x x n ) n 2. T T (K n) (Setting x i = 1 for all i recovers Cayley s formula.)

14 Definitions Counting Spanning Trees Weighted Spanning Tree Enumerators Example: The Cayley-Prüfer Theorem Theorem [Cayley-Prüfer] x T = (x 1 x 2 x n )(x 1 + x x n ) n 2. T T (K n) Combinatorial proof: the Prüfer code, a bijection P : T (K n ) [n] n 2 where deg T (i) = 1 + number of i s in P(T ).

15 Definitions Counting Spanning Trees Weighted Spanning Tree Enumerators More Weighted Spanning Tree Enumerators K p,q : degree sequence (bijection: Hartsfield Werth) Threshold graphs: degree sequence and more (Remmel Williamson) Ferrers graphs: degree sequence (Ehrenborg van Willigenburg) Hypercubes: direction and facet degrees (JLM Reiner; bijection??)

16 Simplicial Complexes Definition A simplicial complex on vertex set V is a family of subsets of V such that {v} for every v V ; If F and G F, then G {{}, 1, 2, 3, 4, 5, 12, 13, 23, 34, 35, 45, 123}

17 Simplicial Complexes Elements of are called faces. Maximal faces are called facets. dim F = F 1; dim = max{dim F F }. is pure if all facets have equal dimension. f i ( ) = number of i-dimensional faces. The k-skeleton is (k) = {F dim F k}. A graph is just a simplicial complex of dimension 1.

18 Simplicial Complexes pure; dim=2 not pure; dim=2 pure; dim=1 f( )=(5,9,7) f( )=(5,6,1) f( )=(5,7)

19 Simplicial Homology R = commutative ring with identity (typically Z or Q) C i ( ) = free R-module on i-dimensional faces of has natural boundary and coboundary maps i : C i C i 1, i : C i 1 C i such that i i+1 = i+1 i = 0.

20 Simplicial Homology Definition The i th reduced simplicial homology group of is H i ( ; R) = ker i / im i+1. Homology groups over Q measure the holes in. Homology groups over Z measure holes (the free part) and twisting (the torsion part). Definition The i th Betti number of is β i ( ) = dim Q H i (, Q).

21 Simplicial Homology Let G be a graph (a 1-dimensional simplicial complex). β0 (G) = (number of connected components of G) 1 β 1 (G) = number of edges that need to be deleted to make G acyclic 1 is the signed vertex-edge incidence matrix M. The Laplacian of G is L = 1 1.

22 Let d be a simplicial complex (i.e., dim = d). Let Υ be a subcomplex with Υ (d 1) = (d 1). Definition Υ is a simplicial spanning tree (SST) of if 1. H d (Υ; Z) = 0; 2. H d 1 (Υ; Q) = 0; 3. f d (Υ) = f d ( ) β d ( ) + β d 1 ( ).

23 Conditions for Υ d to be an SST: 0. Υ (d 1) = (d 1) ( spanning ); 1. Hd (Υ; Z) = 0 ( acyclic ); 2. Hd 1 (Υ; Q) = 0 ( connected ); 3. f d (Υ) = f d ( ) β d ( ) + β d 1 ( ) ( count ). Any two of these conditions together imply the third When d = 1, coincides with the usual definition

24 Metaconnectedness Denote by T ( ) the set of simplicial spanning trees of. Proposition T ( ) if and only if has the homology type of a wedge of d-spheres: β j ( ) = 0 j < dim. Equivalently, H j ( ; Z) < j < dim. Such a complex is called metaconnected.

25 Metaconnectedness metaconnected not metaconnected metaconnected

26 Metaconnectedness Every acyclic complex is metaconnected. Every Cohen-Macaulay complex is metaconnected (by Reisner s theorem), including: 0-dimensional complexes connected graphs simplicial spheres shifted complexes matroid complexes many other complexes arising in algebra and combinatorics

27 Examples of SSTs Example If dim = 0, then T ( ) = {vertices of }. Example If is Q-acyclic, then T ( ) = { }. Includes complexes that are not Z-acyclic, such as RP 2. Example If is a simplicial sphere, then T ( ) = { \ {F } F a facet of }. Simplicial spheres are the analogues of cycle graphs.

28 Kalai s Theorem Graphs and Spanning Trees Let be the d-skeleton of the n-vertex simplex: = {F [n] dim F d}. Theorem [Kalai 1983] Υ T ( ) H d 1 (Υ; Z) 2 = n ( n 2 d ). Reduces to Cayley s formula when d = 1 ( = K n ). Adin (1992): Analogous formula for complete colorful complexes (generalizing Fiedler-Sédlaçek formula for K p,q )

29 Simplicial Analogues of Graph Invariants Let d be a metaconnected simplicial complex. C i 1 ( ) i C i ( ) i C i 1 ( ) L i = i i (the up-down Laplacian) s i = product of nonzero eigenvalues of L i h i = H i 1 (Υ) 2 Υ T ( (i) )

30 The Simplicial Matrix-Tree Theorem Version I s i = product of nonzero eigenvalues of Laplacian L i h i = H i 1 (Υ) 2 Υ T ( (i) ) Theorem [Duval Klivans JLM 26] h d = s d h d 1 H d 2 ( ; Z) 2.

31 Special Cases Graphs and Spanning Trees When is a graph on n vertices, the theorem says that h 1 = s 1 h H 1 ( ) 2 = s 1 0 n which is the classical Matrix-Tree Theorem. If H i (, Z) = 0 for i d 2, then h d = s d s d 2 s d 1 s d 3

32 The Simplicial Matrix-Tree Theorem Version II d = simplicial complex Γ T ( (d 1) ) Γ = restriction of d to faces not in Γ L Γ = Γ Γ Theorem [Duval Klivans JLM 26] h d = Υ T ( ) H d 1 (Υ) 2 = H d 2 ( ; Z) 2 H d 2 (Γ; Z) 2 det L Γ.

33 Theorem (SMTT I: product of eigenvalues) h d = s d h d 1 H d 2 ( ; Z) 2 Theorem (SMTT II: reduced Laplacian) h d = Υ T ( ) H d 1 (Υ) 2 = H d 2 ( ; Z) 2 H d 2 (Γ; Z) 2 det L Γ Version II is more useful for computing h d directly. In many cases, the H d 2 terms are trivial.

34 Introduce an indeterminate x F for each face F Weighted boundary : multiply the F th column of by x F Weighted Laplacian L = Weighted analogues of s i and h i : s i = product of nonzero eigenvalues of L i h i = H i 1 (T ) 2 xf 2 F T T T ( (i) )

35 Weighted Weighted Simplicial Matrix-Tree Theorem I h i = s i h i 1 H i 2 ( ) 2 Weighted Simplicial Matrix-Tree Theorem II where Γ T ( (i 1) ) h i = H i 2 ( ; Z) 2 H i 2 (Γ; Z) 2 det L Γ

36 As in the graphic case, we can use weights to obtain finer enumerative information about simplicial spanning trees. In order for the weighted simplicial spanning tree enumerators to factor, we need L to have integer eigenvalues. That is, must be Laplacian integral. Shifted complexes Matroid complexes Others?

37 Example: The Bipyramid With Equator Vertices: 1, 2, 3, 4, 5 Edges: All but 45 Facets: 123, 124, 134, 234, 125, 135, 235 f ( ) = (5, 9, 7) Equator : the facet 123

38 Example: The Bipyramid With Equator = bipyramid with equator = 123, 124, 134, 234, 125, 135, 235 For each facet F = ijk, set x F = x i x j x k. Enumeration of SSTs of by degree sequence: h 2 = T T ( ) i V x deg T (i) i = x 3 1 x 3 2 x 3 3 x 2 4 x 2 5 (x 1 + x 2 + x 3 )(x 1 + x 2 + x 3 + x 4 + x 5 )

39 A represented matroid is a collection of vectors. A matroid structure on a finite set says, If this were a collection of vectors, then here is a list of which subsets would be linearly independent, bases, etc. If X is a simplicial complex, then the simplicial matroid is the matroid represented by the columns of its boundary map. Simplicial spanning trees are then exactly the bases of this matroid. independence system is... Cell complexes - argument that they are natural for combinatorialists to think about