[PPT] - Arborescences of Derived Graphs CJ Dowd, Sylvester Zhang, Valerie PowerPoint Presentation

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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

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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

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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

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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 Av(Γ) the sum of the weights of all arborescences of Γ rooted at v: Av(Γ) =

T an arborescence

wt(T)

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 2 / 23

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Arborescence Example

1 2 3 b d e c

1 2 3 b e 1 2 3 b d

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 3 / 23

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Arborescence Example

1 2 3 b d e c

1 2 3 b e 1 2 3 b d

A2(Γ) = bd + be

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 3 / 23

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The Laplacian Matrix

Laplacian Matrix: L(Γ) = D(Γ) − A(Γ) Weighted degree matrix: dii =

e=(vi,vj)∈E

wt(e). Adjacency matrix: aij =

e=(vi,vj)∈E

wt(e),

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 4 / 23

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Laplacian Example

1 2 3 b d e c

L(Γ) =   b c d + e   −   b c d e   =   b −b c −c −d −e d + e  

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 5 / 23

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Matrix Tree Theorem

Theorem (Kirchoff)

Given the Laplacian matrix of a graph Γ, Av(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

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Matrix Tree Theorem Example

1 2 3 b d e c L(Γ) =   b −b c −c −d −e d + e  

b

−d d + e

= bd + be

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 7 / 23

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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

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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

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Z/3Z Derived Graph Example

1 2 3 (b, 1) (d, g2) (e, 1) (c, g2) 11 21 31 1g 2g 3g 1g2 2g2 3g2 b b b c c c d d d e e e

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 9 / 23

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Invariance of Arborescence Ratio

Theorem (Galashin–Pylyavskyy, 2017)

If G is simple and strongly connected, then the ratio A˜

v(˜

Γ) Av(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

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The Voltage Laplacian

Voltage Laplacian: L(Γ) = D(Γ) − A(Γ) Weighted degree matrix: dii =

e=(vi,vj)∈E

wt(e). Voltage adjacency matrix: aij =

e=(vi,vj)∈E

ν(e)wt(e),

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 11 / 23

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Voltage Laplacian Example

1 2 3 (b, 1) (d, g2) (e, 1) (c, g2)

L(Γ) =   b c d + e   −   b ζ2

3c

ζ2

3d

e   =   b −b c −ζ2

3c

−ζ2

3d

−e d + e  

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 12 / 23

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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: A˜

v(˜

Γ) Av(Γ) = 1 p

p−1

i=1

det L(Γ, ζi

p)

where L(Γ, ζi) is the voltage Laplacian of Γ evaluated at certain powers of ζp.

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 13 / 23

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Conjecture Example

1 2 3 (b, 1) (d, g2) (e, 1) (c, g2)

1 3

3−1

i=1

det L(Γ, ζi

3) = 1

3 ∗

b

−b c −ζ2

3c

−ζ2

3d

−e d + e

∗
b

−b c −ζ3c −ζ3d −e d + e

= b2c2d2 + b2c2e2 + b2c2ef

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 14 / 23

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Z/2Z case

Theorem (REU 2019)

A˜

v(˜

Γ) Av(Γ) = 1 2 det L(Γ)

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 15 / 23

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Z/2Z case

Theorem (REU 2019)

A˜

v(˜

Γ) Av(Γ) = 1 2 det L(Γ) We’ve proven the special case of the general conjecture when p = 2: A˜

v(˜

Γ) Av(Γ) = 1 p

p−1

i=1

det L(Γ, ζi

p)

Easier to work with as a product identity: 2A˜

v(˜

Γ) = Av(Γ) det L(Γ)

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 15 / 23

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Z/2Z proof by induction sketch

1 2 3 + − − + 1− 2− 3− 1+ 2+ 3+

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

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Z/2Z proof by induction sketch

1 2 3 + − − + 1− 2− 3− 1+ 2+ 3+

Pick root to have ≥ 2 outgoing edges, then partition arborescences of cover into two classes (this step prevents generalization to k > 2, however)

1− 2− 3− 1+ 2+ 3+ 1− 2− 3− 1+ 2+ 3+

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 16 / 23

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Z/2Z proof by induction sketch

Remove other lift of edge as well, since it does not affect aborescences (its initial vertex is the root):

1− 2− 3− 1+ 2+ 3+ 1− 2− 3− 1+ 2+ 3+

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 17 / 23

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Z/2Z proof by induction sketch

Remove other lift of edge as well, since it does not affect aborescences (its initial vertex is the root):

1− 2− 3− 1+ 2+ 3+ 1− 2− 3− 1+ 2+ 3+

We end up with the derived graph of a signed graph with fewer edges:

1 2 3 + − +

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 17 / 23

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Higher covers: progress towards Z/pZ

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

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Higher covers: progress towards Z/pZ

Previous approach does not work; attempt linear algebraic approach by using Matrix Tree Theorem on cover

1− 2− 3− 1+ 2+ 3+

L(˜ Γ) =         a −a b −b c + d −c −d a −a b −b −c −d c + d        

det L3+

3+ = A3+(˜

Γ) = a2b2c + a2b2d

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 18 / 23

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Higher covers: progress towards Z/pZ

Lemma (REU 2019)

Under suitable change of basis, L(˜ Γ) may be written in block matrix form L(Γ) ∗ [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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Higher covers: progress towards Z/pZ

Lemma (REU 2019)

Under suitable change of basis, L(˜ Γ) may be written in block matrix form L(Γ) ∗ [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). We know det[L(Γ)]Q is equal to the norm of det L(Γ), so this is very close to giving us the product formula we want: Av(Γ)NQ(ζp):Q(L(Γ)) = pL˜

v(˜

Γ)

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 19 / 23

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Higher covers: progress towards Z/pZ

Lemma (REU 2019)

Under suitable change of basis, L(˜ Γ) may be written in block matrix form L(Γ) ∗ [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). We know det[L(Γ)]Q is equal to the norm of det L(Γ), so this is very close to giving us the product formula we want: Av(Γ)NQ(ζp):Q(L(Γ)) = pL˜

v(˜

Γ) However, we don’t quite know how to account for taking minors, since change of basis and taking minors do not commute (and we need a factor

f p somewhere).

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 19 / 23

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Higher covers: arbitrary abelian groups

Can always build derived graph by iteratively taking p-fold covers. G = Z/4Z = {1, g, g2, g3}: take two 2-fold covers g g2 − − + −

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 20 / 23

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Higher covers: arbitrary abelian groups

Resulting formula isn’t particularly nice, but (conditioning on our the conjecture for prime cyclic G) we can say

Conjecture (REU 2019)

If Γ is G-volted with G abelian, then the ratio A˜

v(˜ Γ) Av(Γ) is a polynomial in the

edge weights of Γ, and it has positive integer coefficients.

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 21 / 23

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Special thanks

Thank you to Sunita Chepuri, Andy Hardt, Greg Michel, Pasha Pylyavskyy, and Vic Reiner for their advice and mentorship on this problem!

CJ Dowd, Sylvester Zhang, Valerie Zhang (UMN REU 2019) Arborescences of Derived Graphs July 25, 2019 22 / 23

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References

Questions?

S. Chaiken, A combinatorial proof of the All-Minors Matrix Tree

Theorem, SIAM J. Alg Disc. Meth. 3(3), 319-329, 1982.

P. Galashin and P. Pylyavskyy, R-systems, arXiv:1709.00578, 2017.
V. Reiner, D. Tseng, Critical groups of covering, voltage, and signed

graphs, Discrete Mathematics 318, 10-40, 2014.

D. Stanton, D. White, Constructive Combinatorics, 1986.

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