Arborescences of Derived Graphs CJ Dowd, Sylvester Zhang, Valerie Zhang UMN REU 2019 July 25, 2019 CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 1 / 23

Arborescence Definitions Let Γ = ( V , E ) be a directed, edge-weighted graph. CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 2 / 23

Arborescence Definitions Let Γ = ( V , E ) be a directed, edge-weighted graph. An arborescence T of Γ rooted at v ∈ V is a spanning tree directed towards v . The weight of an arborescence wt(T) is the the product of the weights of its edges. CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 2 / 23

Arborescence Definitions Let Γ = ( V , E ) be a directed, edge-weighted graph. An arborescence T of Γ rooted at v ∈ V is a spanning tree directed towards v . The weight of an arborescence wt(T) is the the product of the weights of its edges. We denote by A v (Γ) the sum of the weights of all arborescences of Γ rooted at v : � A v (Γ) = wt ( T ) T an arborescence CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 2 / 23

Arborescence Example 1 d b 1 3 2 d b e 1 3 2 b c e 3 2 CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 3 / 23

Arborescence Example 1 d b 1 3 2 d b e 1 3 2 b c e 3 2 A 2 (Γ) = bd + be CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 3 / 23

The Laplacian Matrix Laplacian Matrix: L (Γ) = D (Γ) − A (Γ) Weighted degree matrix: � d ii = wt( e ) . e =( v i , v j ) ∈ E Adjacency matrix: � a ij = wt( e ) , e =( v i , v j ) ∈ E CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 4 / 23

Laplacian Example 1 d b e 3 2 c b 0 0 0 b 0 − L (Γ) = 0 c 0 0 0 c 0 0 d + e d e 0 b − b 0 = 0 c − c − d − e d + e CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 5 / 23

Matrix Tree Theorem Theorem (Kirchoff) Given the Laplacian matrix of a graph Γ , A v (Γ) is the determinant of the matrix resulting from deleting its corresponding row and column of v. CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 6 / 23

Matrix Tree Theorem Example 1 b − b 0 L (Γ) = 0 c − c − d − e d + e d b e 3 2 � � b 0 c � � � = bd + be � � − d d + e � CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 7 / 23

Voltage Graphs and Derived Graphs A weighted G-voltage graph Γ = ( V , E , wt , ν ) is a directed, edge-weighted graph such that each edge e is also labeled by an element ν ( e ) of a finite group G . This labeling is called a voltage of Γ with respect to G . CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 8 / 23

Voltage Graphs and Derived Graphs A weighted G-voltage graph Γ = ( V , E , wt , ν ) is a directed, edge-weighted graph such that each edge e is also labeled by an element ν ( e ) of a finite group G . This labeling is called a voltage of Γ with respect to G . Given a G -voltage graph Γ, we can construct the derived graph ˜ Γ = ( ˜ V , ˜ E ) where ˜ V := V × G , ˜ E := { [ v × x , w × ( gx )] : x ∈ G , [ v , w ] ∈ E } . CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 8 / 23

Z / 3 Z Derived Graph Example 1 ( d , g 2 ) ( b , 1) ( e , 1) 1 1 3 2 ( c , g 2 ) b e 3 1 2 1 d d 1 g 1 g 2 c c b b d e e 3 g 2 g 3 g 2 2 g 2 c CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 9 / 23

Invariance of Arborescence Ratio Theorem (Galashin–Pylyavskyy, 2017) If G is simple and strongly connected, then the ratio v (˜ A ˜ Γ) A v (Γ) is well-defined and independent of the choice of vertex v and its lift ˜ v. CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 10 / 23

The Voltage Laplacian Voltage Laplacian: L (Γ) = D (Γ) − A (Γ) Weighted degree matrix: � d ii = wt( e ) . e =( v i , v j ) ∈ E Voltage adjacency matrix: � a ij = ν ( e )wt( e ) , e =( v i , v j ) ∈ E CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 11 / 23

Voltage Laplacian Example 1 ( d , g 2 ) ( b , 1) ( e , 1) 3 2 ( c , g 2 ) b 0 0 0 b 0 − ζ 2 L (Γ) = 0 c 0 0 0 3 c ζ 2 0 0 d + e 3 d e 0 b − b 0 − ζ 2 = 0 c 3 c − ζ 2 3 d − e d + e CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 12 / 23

Our Conjecture Conjecture (REU 2019) Let G be a cyclic prime group of order p. Take any vertex v in Γ , and any lift of it in ˜ Γ , say ˜ v, then the following is true: p − 1 v (˜ A v (Γ) = 1 A ˜ Γ) � det L (Γ , ζ i p ) p i =1 where L (Γ , ζ i ) is the voltage Laplacian of Γ evaluated at certain powers of ζ p . CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 13 / 23

Conjecture Example 1 ( d , g 2 ) ( b , 1) ( e , 1) 3 2 ( c , g 2 ) � � � � b − b 0 b − b 0 3 − 1 � � � � 1 3 ) = 1 � det L (Γ , ζ i � − ζ 2 � � � 3 ∗ 0 c 3 c ∗ 0 c − ζ 3 c � � � � 3 � − ζ 2 � � � 3 d − e d + e − ζ 3 d − e d + e i =1 � � � � = b 2 c 2 d 2 + b 2 c 2 e 2 + b 2 c 2 ef CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 14 / 23

Z / 2 Z case Theorem (REU 2019) v (˜ A ˜ A v (Γ) = 1 Γ) 2 det L (Γ) CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 15 / 23

Z / 2 Z case Theorem (REU 2019) v (˜ A ˜ A v (Γ) = 1 Γ) 2 det L (Γ) We’ve proven the special case of the general conjecture when p = 2: p − 1 v (˜ A v (Γ) = 1 A ˜ Γ) � det L (Γ , ζ i p ) p i =1 Easier to work with as a product identity: v (˜ 2 A ˜ Γ) = A v (Γ) det L (Γ) CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 15 / 23

Z / 2 Z proof by induction sketch 1 + 1 1 − − + − 2 − 3 2 3 − + 3 + 2 + Pick root to have ≥ 2 outgoing edges, then partition arborescences of cover into two classes (this step prevents generalization to k > 2, however) CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 16 / 23

Z / 2 Z proof by induction sketch 1 + 1 1 − − + − 2 − 3 2 3 − + 3 + 2 + Pick root to have ≥ 2 outgoing edges, then partition arborescences of cover into two classes (this step prevents generalization to k > 2, however) 1 + 1 + 1 − 1 − 2 − 2 − 3 − 3 − 3 + 2 + 3 + 2 + CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 16 / 23

Z / 2 Z proof by induction sketch Remove other lift of edge as well, since it does not affect aborescences (its initial vertex is the root): 1 + 1 + 1 − 1 − 2 − 2 − 3 − 3 − 3 + 2 + 3 + 2 + CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 17 / 23

Z / 2 Z proof by induction sketch Remove other lift of edge as well, since it does not affect aborescences (its initial vertex is the root): 1 + 1 + 1 − 1 − 2 − 2 − 3 − 3 − 3 + 2 + 3 + 2 + We end up with the derived graph of a signed graph with fewer edges: 1 + − 3 2 + CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 17 / 23

Higher covers: progress towards Z / p Z Previous approach does not work; attempt linear algebraic approach by using Matrix Tree Theorem on cover CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 18 / 23

Higher covers: progress towards Z / p Z Previous approach does not work; attempt linear algebraic approach by using Matrix Tree Theorem on cover 1 + 1 − a − a 0 0 0 0 0 b − b 0 0 0 2 − 3 − 0 0 c + d − c − d 0 L (˜ Γ) = 3 + 2 + 0 0 0 a − a 0 0 0 0 0 b − b − c − d 0 0 0 c + d det L 3 + 3 + = A 3 + (˜ Γ) = a 2 b 2 c + a 2 b 2 d CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 18 / 23

Higher covers: progress towards Z / p Z Lemma (REU 2019) Under suitable change of basis, L (˜ Γ) may be written in block matrix form � L (Γ) � ∗ 0 [ L (Γ)] Q where L (Γ) is the ordinary Laplacian matrix of Γ and [ L (Γ)] Q is the voltage Laplacian of Γ written as a matrix with entries in Q (restriction of scalars). CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 19 / 23

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