Counting the spanning trees of the 3-cube using edge slides Christopher Tuffley Institute of Fundamental Sciences Massey University, Manawatu 2009 New Zealand Mathematics Colloquium Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 1 / 15
Outline Introduction 1 Cubes and spanning trees Counting spanning trees: ways and means The 3-cube 2 Edge slides Counting the trees Higher dimensions 3 Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 2 / 15
Introduction Cubes and spanning trees Cubes { 1 , 2 , 3 } Definition The n -cube is the graph Q n with { 1 , 2 } { 2 , 3 } { 1 , 3 } vertices the subsets of [ n ] = { 1 , 2 , . . . , n } ; { 2 } an edge between S and R if they { 1 } { 3 } differ by adding or deleting a single element. ∅ Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 3 / 15
Introduction Cubes and spanning trees Spanning trees Definition A spanning tree of a connected graph G is a maximal subset of the edges that contains no cycle; equivalently, a minimal subset of the edges that connects all the vertices. Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 4 / 15
Introduction Counting spanning trees: ways and means Counting trees — the matrix way Theorem (Kirchoff’s Matrix-Tree Theorem) The number of spanning trees of a simple connected graph G is given by the determinant of a matrix associated with G — the Laplacian of G, with row i, column i deleted. 1 4 3 − 1 − 1 − 1 − 1 2 − 1 0 − 1 − 1 2 0 − 1 0 0 1 2 3 Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 5 / 15
Introduction Counting spanning trees: ways and means Counting trees — the matrix way Theorem (Kirchoff’s Matrix-Tree Theorem) The number of spanning trees of a simple connected graph G is given by the determinant of a matrix associated with G — the Laplacian of G, with row i, column i deleted. 1 4 2 − 1 0 det = 3 − 1 2 0 0 0 1 2 3 Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 5 / 15
Introduction Counting spanning trees: ways and means Counting trees — the combinatorial way Model: Prüfer code for spanning trees of K n (Prüfer, 1918) The Prüfer code is a bijection → { 1 , . . . , n } n − 2 spanning trees of K n ← — recovering Cayley’s Theorem that K n has n n − 2 spanning trees. 1 2 6 Prüfer code 3411 5 3 4 Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 6 / 15
Introduction Counting spanning trees: ways and means Spanning trees of the n -cube Known result The n-cube has n k ( n 2 | S | = 2 2 n − n − 1 k ) � � S ⊆ [ n ] k = 1 | S |≥ 2 spanning trees. For n = 3 this gives 2 4 · 2 3 · 3 = 384 spanning trees. Proof. The Matrix-Tree Theorem + clever determination of eigenvalues. See e.g. Stanley, Enumerative Combinatorics , Vol II. Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 7 / 15
Introduction Counting spanning trees: ways and means Spanning trees of the n -cube Known result The n-cube has n k ( n 2 | S | = 2 2 n − n − 1 k ) � � S ⊆ [ n ] k = 1 | S |≥ 2 spanning trees. For n = 3 this gives 2 4 · 2 3 · 3 = 384 spanning trees. Proof. The Matrix-Tree Theorem + clever determination of eigenvalues. See e.g. Stanley, Enumerative Combinatorics , Vol II. Problem Stanley: “A direct combinatorial proof of this formula is not known.” Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 7 / 15
Introduction Counting spanning trees: ways and means A weighted count Theorem (Martin and Reiner, 2003) With respect to certain weights q 1 , . . . , q n , x 1 , . . . , x n we have q dir ( T ) x dd ( T ) = q 1 · · · q n � � � q i ( x − 1 + x i ) . i s. trees S ⊆ [ n ] i ∈ S | S |≥ 2 of Q n degree of q i in q dir ( T ) is the number of edges in direction i degree of x i in x dd ( T ) is the number of edges in the “upper” i -face minus the number in the “lower”. Suggests that a spanning tree of Q n � a choice of element and sign at each vertex of cardinality 2. Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 8 / 15
Introduction Counting spanning trees: ways and means A weighted count Theorem (Martin and Reiner, 2003) With respect to certain weights q 1 , . . . , q n , x 1 , . . . , x n we have q dir ( T ) x dd ( T ) = q 1 · · · q n � � � q i ( x − 1 + x i ) . i s. trees S ⊆ [ n ] i ∈ S | S |≥ 2 of Q n degree of q i in q dir ( T ) is the number of edges in direction i degree of x i in x dd ( T ) is the number of edges in the “upper” i -face minus the number in the “lower”. Suggests that a spanning tree of Q n � a choice of element and sign at each vertex of cardinality 2. Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 8 / 15
The 3-cube Edge slides Edge slides { 1 , 2 , 3 } Definition An edge of a spanning tree is slidable { 1 , 2 } { 2 , 3 } if it can be “slid” across a face of the { 1 , 3 } cube to give a second spanning tree. Observation { 1 } { 3 } { 2 } An edge that may be slid in direction i must lie on the path joining two i-edges. ∅ Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 9 / 15
The 3-cube Edge slides Edge slides { 1 , 2 , 3 } Definition An edge of a spanning tree is slidable { 1 , 2 } { 2 , 3 } if it can be “slid” across a face of the { 1 , 3 } cube to give a second spanning tree. Observation { 1 } { 3 } { 2 } An edge that may be slid in direction i must lie on the path joining two i-edges. ∅ Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 9 / 15
The 3-cube Edge slides Edge slides { 1 , 2 , 3 } Definition An edge of a spanning tree is slidable { 1 , 2 } { 2 , 3 } if it can be “slid” across a face of the { 1 , 3 } cube to give a second spanning tree. Observation { 1 } { 3 } { 2 } An edge that may be slid in direction i must lie on the path joining two i-edges. ∅ Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 9 / 15
The 3-cube Edge slides Edge slides { 1 , 2 , 3 } Definition An edge of a spanning tree is slidable { 1 , 2 } { 2 , 3 } if it can be “slid” across a face of the { 1 , 3 } cube to give a second spanning tree. Observation { 1 } { 3 } { 2 } An edge that may be slid in direction i must lie on the path joining two i-edges. ∅ Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 9 / 15
The 3-cube Edge slides Edge slides { 1 , 2 , 3 } Definition An edge of a spanning tree is slidable { 1 , 2 } { 2 , 3 } if it can be “slid” across a face of the { 1 , 3 } cube to give a second spanning tree. Observation { 1 } { 3 } { 2 } An edge that may be slid in direction i must lie on the path joining two i-edges. ∅ Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 9 / 15
The 3-cube Edge slides Existence Lemma A minimal path joining two i-edges contains a unique edge that may be slid in direction i. Proof (length three case only). u v Vertices u and v must meet edges of the tree. There are three possibilities. Corollary A tree with k edges in direction i has k − 1 edges that may be slid in direction i, for a total of exactly four possible slides. Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 10 / 15
The 3-cube Edge slides Existence Lemma A minimal path joining two i-edges contains a unique edge that may be slid in direction i. Proof (length three case only). u v Vertices u and v must meet edges of the tree. There are three possibilities. Corollary A tree with k edges in direction i has k − 1 edges that may be slid in direction i, for a total of exactly four possible slides. Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 10 / 15
The 3-cube Edge slides Existence Lemma A minimal path joining two i-edges contains a unique edge that may be slid in direction i. Proof (length three case only). u v Vertices u and v must meet edges of the tree. There are three possibilities. Corollary A tree with k edges in direction i has k − 1 edges that may be slid in direction i, for a total of exactly four possible slides. Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 10 / 15
The 3-cube Edge slides Existence Lemma A minimal path joining two i-edges contains a unique edge that may be slid in direction i. Proof (length three case only). u v Vertices u and v must meet edges of the tree. There are three possibilities. Corollary A tree with k edges in direction i has k − 1 edges that may be slid in direction i, for a total of exactly four possible slides. Christopher Tuffley (Massey University) Counting the spanning trees of the 3-cube NZMC 2009 10 / 15
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