Spanning trees of tree graphs Philippe Biane, CNRS-IGM-Universit´ e Paris-Est Firenze, May 18 2015 joint work with Guillaume Chapuy, CNRS-LIAFA-Universit´ e Paris 7 Philippe Biane spanning trees of tree graphs
V , E =directed graph x e w v Q =Laplacian matrix, indexed by V × V ◮ Q vw = x e if e : v → w is a directed edge of the graph ◮ Q vv = − � w Q vw Philippe Biane spanning trees of tree graphs
Markov chains If x e ≥ 0 this is the generator of a continuous time Markov chain on the graph, with transition probabilities e tQ . ◮ Q 1 = 0 where 1 is the constant vector. ◮ If the chain is irreducible or the graph is strongly connected the kernel is one dimensional (Perron Frobenius). ◮ µ Q = 0 for a unique positive invariant measure µ . Philippe Biane spanning trees of tree graphs
Rooted spanning trees v A spanning tree rooted at v Philippe Biane spanning trees of tree graphs
Kirchhoff’theorem X ⊂ V Q X = Q with rows and columns in X � � det( Q X ) = x e f ∈ F X e ∈ f The sum is over forests rooted in V \ X . In particular the invariant measure is � � µ ( v ) = x e e ∈ t t ∈ T v sum over oriented trees rooted at v Philippe Biane spanning trees of tree graphs
Tree-graph of a graph TG = ( TV , TE ) TV =Vertices of the tree graph=spanning trees of the graph s =spanning tree rooted at v e = v → w edge from s to t : w v The tree s Philippe Biane spanning trees of tree graphs
Tree-graph of a graph TG = ( TV , TE ) TV =Vertices of the tree graph=spanning trees of the graph s =spanning tree rooted at v e = v → w edge from s to t : w v Philippe Biane spanning trees of tree graphs
Tree-graph of a graph TG = ( TV , TE ) TV =Vertices of the tree graph=spanning trees of the graph s =spanning tree rooted at v e = v → w edge from s to t : w v The tree t Philippe Biane spanning trees of tree graphs
The tree-graph is a covering graph: p : s �→ v mapping each tree to its root. Every path in V can be lifted to TV . Philippe Biane spanning trees of tree graphs
Example { 3 , 4 } { 1 , 2 , 3 , 4 } { 4 } x 23 { 1 , 3 , 4 } { 1 , 2 , 3 , 4 } x 31 { 3 , 4 } { 1 , 3 , 4 } 1 { 3 , 4 } x 43 { 1 , 2 , 3 , 4 } x 34 2 { 4 } 3 4 { 1 , 2 , 3 , 4 } { 4 } { 1 , 3 , 4 } { 3 , 4 } Philippe Biane spanning trees of tree graphs
Laplacian matrix of the tree-graph On each edge s → t above v → w put the weight x e . This defines the Laplacian matrix R of the tree-graph R st = Q vw ; p ( s ) = v ; p ( t ) = w This is the generator of a continuous time Markov chain on the tree-graph. Philippe Biane spanning trees of tree graphs
Lifting of the Markov chain The chain on TV projects to the chain on V by p : TV → V : if TX is a R -Markov chain on TV then p ( TX ) is a Q -Markov chain on V . { 1 , 2 , 3 , 4 } { 3 , 4 } { 4 } x 23 { 1 , 3 , 4 } { 1 , 2 , 3 , 4 } x 31 { 1 , 3 , 4 } { 3 , 4 } 1 { 3 , 4 } x 43 { 1 , 2 , 3 , 4 } x 34 2 { 4 } 3 4 { 1 , 2 , 3 , 4 } { 1 , 3 , 4 } { 4 } { 3 , 4 } Philippe Biane spanning trees of tree graphs
Lemma: the invariant measure of the chain on the tree graph is � T µ ( t ) = x e e ∈ t This provides a combinatorial proof of Kirchhoff’s theorem since p ( T µ ) = µ � � � µ ( v ) = T µ ( t ) = x e e ∈ t t ∈ p − 1 ( v ) t ∈ T v Philippe Biane spanning trees of tree graphs
Spanning trees of the tree-graph The invariant measure of the chain on the tree graph can also be computed using spanning trees of the tree graph. The preceding result implies � � � x e = P ( x e ; e ∈ E ) x e t ∈ T t e ∈ t e ∈ t the sum is over spanning trees of TG rooted at t . The polynomial P is independent of t , it depends only on the graph V . Philippe Biane spanning trees of tree graphs
Example The complete graph on X = { 1 , 2 , 3 } 2 u v a b c 1 3 w λ a w Q = u µ b c v ν with λ = − a − w , µ = − b − u , ν = − c − v Philippe Biane spanning trees of tree graphs
vw 2 v b bw 3 c w u v bc 1 c a b ab 3 ac 2 a b v u av 2 cu 1 a c u w w uv 1 uw 3 Figure : The graph T Philippe Biane spanning trees of tree graphs
The transition matrix for the lifted Markov chain is λ 0 0 0 a 0 w 0 0 0 0 a 0 0 w 0 0 λ 0 0 λ 0 a 0 0 w 0 0 u 0 µ 0 0 0 0 b R = 0 0 0 0 0 0 u µ b 0 u 0 0 0 µ 0 b 0 c 0 0 0 0 v 0 0 ν 0 0 0 0 0 0 c v ν 0 0 c v 0 0 0 0 ν Philippe Biane spanning trees of tree graphs
The polynomial P can be computed P ( a , b , c , u , v , w ) = ( bc + cu + uv )( av + ac + vw )( ab + bw + uw ) � � = π ( t ) i ∈ X t ∈ T i It is a product of the 2-minors of the matrix λ a w Q = u µ b c v ν λ = − a − w , µ = − b − u , ν = − c − v Philippe Biane spanning trees of tree graphs
Theorem There exist integers m ( W ); W ⊂ V such that � det( Q W ) m ( W ) P ( x e ; e ∈ E ) = W � V Philippe Biane spanning trees of tree graphs
Computation of the multiplicities m ( W ) Fix a total ordering of the vertex set V of G. Start with a vertex v , and a spanning tree t rooted at v . Perform breadth first search of the graph t and for each vertex obtained erase it if the edge is not in the tree t . 1 8 9 2 7 3 4 5 6 This yields a tree on vertex set X . The output is the strongly connected component of v in X . Philippe Biane spanning trees of tree graphs
Computation of the multiplicities m ( W ) Fix a total ordering of the vertex set V of G. Start with a vertex v , and a spanning tree t rooted at v . Perform breadth first search of the graph t and for each vertex obtained erase it if the edge is not in the tree t . 1 8 9 2 7 3 4 5 6 This yields a tree on vertex set X . The output is the strongly connected component of v in X . Philippe Biane spanning trees of tree graphs
Computation of the multiplicities m ( W ) Fix a total ordering of the vertex set V of G. Start with a vertex v , and a spanning tree t rooted at v . Perform breadth first search of the graph t and for each vertex obtained erase it if the edge is not in the tree t . 1 8 9 2 7 3 4 5 6 This yields a tree on vertex set X . The output is the strongly connected component of v in X . Philippe Biane spanning trees of tree graphs
Computation of the multiplicities m ( W ) Fix a total ordering of the vertex set V of G. Start with a vertex v , and a spanning tree t rooted at v . Perform breadth first search of the graph t and for each vertex obtained erase it if the edge is not in the tree t . 1 8 9 7 3 4 5 6 This yields a tree on vertex set X . The output is the strongly connected component of v in X . Philippe Biane spanning trees of tree graphs
Computation of the multiplicities m ( W ) Fix a total ordering of the vertex set V of G. Start with a vertex v , and a spanning tree t rooted at v . Perform breadth first search of the graph t and for each vertex obtained erase it if the edge is not in the tree t . 1 8 9 7 3 4 5 6 This yields a tree on vertex set X . The output is the strongly connected component of v in X . Philippe Biane spanning trees of tree graphs
Computation of the multiplicities m ( W ) Fix a total ordering of the vertex set V of G. Start with a vertex v , and a spanning tree t rooted at v . Perform breadth first search of the graph t and for each vertex obtained erase it if the edge is not in the tree t . 1 8 9 7 3 4 5 6 This yields a tree on vertex set X . The output is the strongly connected component of v in X . Philippe Biane spanning trees of tree graphs
Computation of the multiplicities m ( W ) Fix a total ordering of the vertex set V of G. Start with a vertex v , and a spanning tree t rooted at v . Perform breadth first search of the graph t and for each vertex obtained erase it if the edge is not in the tree t . 1 9 7 3 4 5 6 This yields a tree on vertex set X . The output is the strongly connected component of v in X . Philippe Biane spanning trees of tree graphs
Computation of the multiplicities m ( W ) Fix a total ordering of the vertex set V of G. Start with a vertex v , and a spanning tree t rooted at v . Perform breadth first search of the graph t and for each vertex obtained erase it if the edge is not in the tree t . 1 9 7 3 4 5 6 This yields a tree on vertex set X . The output is the strongly connected component of v in X . Philippe Biane spanning trees of tree graphs
Computation of the multiplicities m ( W ) Fix a total ordering of the vertex set V of G. Start with a vertex v , and a spanning tree t rooted at v . Perform breadth first search of the graph t and for each vertex obtained erase it if the edge is not in the tree t . 9 7 3 4 5 6 This yields a tree on vertex set X . The output is the strongly connected component of v in X . Philippe Biane spanning trees of tree graphs
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