HW 1, due Friday, September 3
Math 412
I learned about the following, leading to a characterization of the elements
of the sandpile group, from exercises for an REU led by L`szl` Babai.
ao
A semigroup is a set with an associative binary operation. An idea
WK rer slim/J Swim. SAm a1l JILL mm 4: A Jva
HAL am «Star, 13} m fix Jwa C13 M 6? L.
19:" M,).--)M,H LR PM) a; A OCHW WMWS (Ulkvxyx
0% Lch e,\,.--,Q,\ 2: Hug shng Ems makers ¥w Z5
Wt. qux M
M /\ -- NAM! Am) C\/\"'/\ m.
. , - . Ml
PECVI 0w) LNI
HW 2, due Friday, September 10
Math 412
1. (a) Show that every nite abelian group is the sandpile group of some
graph.
(b) Find two non-isomorphic graphs with the same sandpile group.
(An isomorphism of graphs is a bijection of vertex sets inducing a
bije
Math 412
HW 3, due Friday, September 17
1. Let be the following digraph:
v1
2
2
v3
v2
s
(a) Let be the reduced Laplacian of with respect to its global
sink. By hand, nd 3 3 integer matrices, U, V , invertible over
the integers, such that U V = D where D i
Math 412
HW 4, due Friday, September 24
1. Adding a source vertex. Let (, s) be a sandpile graph.
(a) Let k 1 and choose v1 , . . . , vk V . Form the digraph obtained from by adding a (source) vertex, u, and directed edges
(u, vi ) for i = 1, . . . , k .
HW 5, due Friday, October 1
Math 412
Let be a connected, undirected (nite) graph with genus g and canonical
divisor K . Let D be a divisor on .
1. Describe all graphs with genus 0.
2. Describe a family of graphs cfw_i i0 for which the genus of i is i.
3.
Math 412
HW 6, due Friday, October 8
1. Let T be a triangle with vertices cfw_0, 1, 2 but with the edge from 1 to 2
having weight 3. Take 0 as the sink.
3
1
2
0
(a) Find all the recurrents and all the superstables on T .
(b) For each recurrent c, nd 0 suc
Math 412
HW 7, due Friday, October 29
1. Let be a directed multigraph.
(a) Show that D div() is alive i |Dmax D| = .
(b) Dene the nonspecial divisors, N , to be the maximal (with respect to component-wise comparision) divisors with empty linear
system. De
Math 412
HW 8, due Friday, November 5
1. Riemann-Roch for directed graphs.
(a) Suppose is a directed multigraph with trivial critical group.
Show there is a unique divisor class K and a unique integer g
such that
r(D) r(K D) = deg D + 1 g
for all divisors
Mank 412 HM Q)
I. a) cum +9.1:R =7?!K<A:V(P3WX $367; s,'\.
r693 ' mph) : AU Dew/3 V hm (F).
PW mg Dso, av 1m *0 Wm M r(03r(\\5(j)
7.e«) C DzK, M WA n(\i\' rCcclK-},
Le.) :MKth, M0 on} K:
Sana. JV: 0?) Hm,
MD) = L H 33" 0
LSD 9? $3530.
for %D37
Math 412
Matrix-Tree Theorem
Denition 1 Let be a directed graph, and let s be a vertex of . A
spanning tree of directed into s, also called a spanning tree rooted at s, is
a connected subgraph of containing all of the vertices and such that each
vertex ha