Polyhedral Constructions using the Laplacian Matrix
Marie Meyer
Department of Mathematics
University of Kentucky
January 19, 2017
Marie Meyer (University of Kentucky)
Polyhedral Constructions using the Laplacian Matrix
January 19, 2017
1 / 10
Outline
1
Proof Sketch of the Matrix Tree Theorem
Definitions and Notation
2
Dall and Pfeifle’s Polyhedral Construction
A Polyhedral Proof of the Matrix Tree Theorem (2014)
3
Currenct Polyhedral Construction
Applying Ehrhart Theory
Marie Meyer (University of Kentucky)
Polyhedral Constructions using the Laplacian Matrix
January 19, 2017
2 / 10
Definitions
G := ([n], E ) is a connected
graph with n vertices and
|E | = d.
N := signed vertex-edge
incidence matrix is the
(n × d)-matrix of rank n − 1.
L := Laplacian matrix is the
n × n matrix L = N · N T with
eigenvalues
{0, λ1 , λ2 , · · · , λn−1 }, λi ∈ R>0 .
Marie Meyer (University of Kentucky)


−1 0
N =  1 −1
0
1


1 −1 0
L = −1 2 −1
0 −1 1
Polyhedral Constructions using the Laplacian Matrix
January 19, 2017
3 / 10
Definition
The zonotope generated by an (n × d) - matrix M of rank r is the
Minkowski sum of the line segments conv{0, Mi }, where Mi is the ith
column of M.
d
X
Z (M) = {
αi Mi |αi ∈ [0, 1]}
i=1
Marie Meyer (University of Kentucky)
Polyhedral Constructions using the Laplacian Matrix
January 19, 2017
4 / 10
Definition
The zonotope generated by an (n × d) - matrix M of rank r is the
Minkowski sum of the line segments conv{0, Mi }, where Mi is the ith
column of M.
d
X
Z (M) = {
αi Mi |αi ∈ [0, 1]}
i=1
Theorem (Stanley)
Z (M) is the (almost) disjoint union of parallelepipeds indexed by linearly
independent columns of M.
Marie Meyer (University of Kentucky)
Polyhedral Constructions using the Laplacian Matrix
January 19, 2017
4 / 10
Proof Sketch
Marie Meyer (University of Kentucky)
Polyhedral Constructions using the Laplacian Matrix
January 19, 2017
5 / 10
Proof Sketch
Marie Meyer (University of Kentucky)
Polyhedral Constructions using the Laplacian Matrix
January 19, 2017
6 / 10
Proof Sketch
Marie Meyer (University of Kentucky)
Polyhedral Constructions using the Laplacian Matrix
January 19, 2017
7 / 10
Proof Sketch
Marie Meyer (University of Kentucky)
Polyhedral Constructions using the Laplacian Matrix
January 19, 2017
8 / 10
Proof Sketch
Marie Meyer (University of Kentucky)
Polyhedral Constructions using the Laplacian Matrix
January 19, 2017
9 / 10
Thanks for listening!
Marie Meyer (University of Kentucky)
Polyhedral Constructions using the Laplacian Matrix
January 19, 2017
10 / 10