G-Parking Functions, Graph
Searching, and Tutte Polynomial
Huafei Yan
Nankai University andTexas A&M University
Joint with Dimitrije Kostic
1. BFS on a connected graphs
Start a queue which is initially {0}. At
each stage we take the vertex x at the
head of the queue, remove x from the
queue, and add all new neighbors of x
to the queue.
--- Spencer: Enumerating Graphs and Brownian Motion,
(1997)
BFS on H
0
3
4
2
1
5
BFS on H
0
t
3
4
2
1
5
0
Queue
0
BFS on H
0
t
3
4
2
1
5
Queue
0
0
1
3,4
BFS on H
0
t
3
4
2
1
5
Queue
0
0
1
3,4
2
4,1,2
BFS on H
0
t
3
4
2
1
5
Queue
0
0
1
3,4
2
4,1,2
3
1,2
BFS on H
0
t
3
4
2
1
5
Queue
0
0
1
3,4
2
4,1,2
3
1,2
4
2,5
BFS on H
0
t
3
4
2
1
5
Queue
0
0
1
3,4
2
4,1,2
3
1,2
4
2,5
5
5
6
--
Which H s.t. BFS(H)=T?
0
t
3
4
2
1
5
Queue
0
0
1
3,4
2
4,1,2
3
1,2
4
2,5
5
5
6
--

[Spencer] An edge {u,v} can be added to
T iff u and v have been present in the
queue at the same time.
Ex(T)=set of such edges.
Theorem. Given T.
BFS(H)=T iff H  [T, T  Ex(T) ] .
2. A familiar statistics

Let M(T)=|Ex(T)|. Then number of
labeled connected graphs on n+1
vertices with n+k edges is
Let
Cn(q)=G q|E(G)|-n,
G: labeled, connected, n+1 vertices.
Mn(q)=T q|Ex(T)|,
Then
Cn(q) = Mn(1+q).
Same property holds for
 external activity of trees
 inversion of trees
 level in recurrent configurations of sandpile
model
and
 (reversed) sum of parking functions…..
Parking function
A PF is a sequence (a1,a2,…, an) such
that the number of terms larger than k is
less than n-k.
n=1. (0 )
 n=2. (0,0), (0,1), (1,0)
 There are (n+1)n-1 many parking functions of
length n.

Reversed sum of PFs

Let a=(a1,a2,…, an) be a PF. The
reversed sum rsum(a) is
i (i-1-ai) = n(n-1)/2-i ai
Reversed sum of PFs

Let a=(a1,a2,…, an) be a PF. The
reversed sum rsum(a) is
i (i-1-ai) = n(n-1)/2-i ai
rsum(a) has the same distribution
as M(T).
3. PF as a vertex function
G=Kn+1 with vertex set {0,1,…,n}

A PF is a function from {1,2,…,n} to nonnegative integers with the property:
For each nonempty subset U of
{1,2,…,n}, there is a vertex v in U s.t.
a(v) < n-|U|.
An example
0
t
3/
4/
2/
1/
5/
Queue
0
0
1
3,4
2
4,1,2
3
1,2
4
2,5
5
5
6
--
a(v) = rank of the parent of v
0
t
3/ 0
4/ 0
2/ 1
1/ 1
5/ 3
Queue
0
0
1
3,4
2
4,1,2
3
1,2
4
2,5
5
5
6
--
a(v) = rank of the parent of v
0
3/ 0
4/ 0
2/ 1
1/ 1
5/ 3
M(T)
t
Q
0
0
1
3,4
2
4,1,2
3
1,2
4
2,5
5
5
6
--
rsum(a)
4.G-parking functions
Definition. A G-parking function is a function f
from {1,2,…,n} to non-negative integers with
the property:
For each nonempty subset U of {1,2,…,n},
there is a vertex v in U s.t. the number of
edges from v to vertices outside of U is
greater than f(v).
0 0
4/ 1
1 1/ 0
2/ 2
234
34
3
3/ 2
Tutte polynomial of G

To count connected subgraphs of G by
the number of excess edges, use
Tutte polynomial tG(x,y)
Theorem.
tG(1+x,1+y) = H xc(H)-1 y|E(H)|+c(H)-n-1
where H is over all spanning
subgraphs.
General picture
G-parking functions
Tutte polynomial of G
bijections
BFS
Spanning trees of G
5. BFS to subgraphs of G
Theorem. Given G and a spanning tree T.
Then BFS(H)=T iff
H 2 [T, T[ (Ex(T) \ G) ]
Corollary.
tG(1, y) = T y|Ex(T) in
G|
where T ranges over all spanning trees of G.
BFS to subgraphs of G
Theorem. Given G and a spanning tree T. Then
BFS(H)=T iff
H 2 [T, T[ (Ex(T) \ G) ]
Corollary.
tG(1, y) = T y|Ex(T) in G|
where T ranges over all spanning trees of
G.
6. From T to G-parking function
Given T in G, apply BFS on T.
Define
f(v) = number of edges {w,v} in G such
that w is processed before the parent
of v in the queue.

An example
0
3
4
1
2
5
An example
0
t
3/
4/
2/
1/
5/
Queue
0
0
1
3,4
2
4,1,2
3
1,2
4
2,5
5
5
6
--
f(v)={ (u,v) in E(G):
rank(u)<rank(parent of v) }
0
t
3/ 0
4/ 0
2/ 1
1/ 0
5/ 2
Queue
0
0
1
3,4
2
4,1,2
3
1,2
4
2,5
5
5
6
--
7.From G-parking function to tree
BFS with a value function.

Initially, val_0(v)=f(v)

Run BFS on G and update the value
function

At each stage, add new neighbors only if
the value is -1.
Example
0
t
3/ 0
4/ 0
2/ 1
1/ 0
5/ 2
0
Queue
0
Example
0
3/ -1
4/ -1
1/ 0
2/ 0
5/ 1
t
Q
0
0
1
3,4
Example
0
3/-1
4/ -2
2/ -1
1/ -1
5/ 1
t
Q
0
0
1
3,4
2
4,1,2
Example
0
3/-1
4/ -2
2/ -2
1/ -2
5/ 0
t
Q
0
0
1
3,4
2
4,1,2
3
1,2
Example
0
3/-1
4/ -2
1/ -2
2/ -3
5/ -1
t
Q
0
0
1
3,4
2
4,1,2
3
1,2
4
2,5
Example
0
3/-1
4/ -2
2/ -3
1/ -2
5/ -2
t
Q
0
0
1
3,4
2
4,1,2
3
1,2
4
2,5
5
5
Example
0
3/-1
4/ -2
2/ -3
1/ -2
5/ -2
t
Q
0
0
1
3,4
2
4,1,2
3
1,2
4
2,5
5
6
5
--


The ending value records the number of
“extra edges”.
|E(G)|= v f(v) +|E(T)| +|Ex(T) in G|
Example
0
3/-1
4/ -2
2/ -3
1/ -2
5/ -2
t
Q
0
0
1
3,4
2
4,1,2
3
1,2
4
2,5
5
6
5
--
Conclusion
Let rsum(f) = |E(G)|-n-v f(v)
Theorem.
tG(1,y) = f yrsum(f)
One can get the full tG(x,y) by allowing multiroots for the
G-parking function.
8 Multiparking functions
Definition. A G-multiparking function is a
function f from {1,2,…,n} to non-negative
integers and (*) with the property:
For each nonempty subset U of {1,2,…,n}, either
(i) f(v)=* where v=min(U), or (ii) there is a vertex
v in U s.t. the number of edges from v to vertices
outside of U is greater than f(v).
General formula

Let r(f)=number of v s.t. f(v)=*
Theorem.
tG(1+x,y) = yE(G)-n+1f (xy)r(f)-1y-sum(f)-Rec(f) ,
where Rec(f) is the number of edges incident to roots.
A corollary

In a parking function (a1a2…an), a term
ai=j is critical if
(i) no other term =j;
(ii) There are j terms <j, and n-j-1 terms >j.
Let p(a1…an) = #{ j: j is critical, and a leftto-right maximal}
Theorem
TKn+1(x,y)=f 2 P(n) xp(f)yn(n-1)/2-sum(f).
Example: n=2
(0,1),
(1,0),
TK3 = x2 + x + y.
(0,0)