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G-Parking Functions, Graph Searching, and Tutte Polynomial
Huafei YanNankai 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 --- Spencer: Enumerating Graphs and Brownian MotionMotion, (1997)
BFS on H
0
4
2
3
5
1
BFS on H
0
4
2
3
5
1
t Queue
0 0
BFS on H
0
4
2
3
5
1
t Queue
0 0
1 3,4
BFS on H
0
4
2
3
5
1
t Queue
0 0
1 3,4
2 4,1,2
BFS on H
0
4
2
3
5
1
t Queue
0 0
1 3,4
2 4,1,2
3 1,2
BFS on H
0
4
2
3
5
1
t Queue
0 0
1 3,4
2 4,1,2
3 1,2
4 2,5
BFS on H
0
4
2
3
5
1
t 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
4
2
3
5
1
t 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).
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 non-negative 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
4/
2/
5/
1/
t Queue
0 0
1 3,4
2 4,1,2
3 1,2
4 2,5
5 5
6 --
3/
a(v) = rank of the parent of v
0
4/ 0
2/ 1
5/ 3
1/ 1
t Queue
0 0
1 3,4
2 4,1,2
3 1,2
4 2,5
5 5
6 --
3/ 0
a(v) = rank of the parent of v
0
4/ 0
2/ 1
5/ 3
1/ 1
t Q
0 0
1 3,4
2 4,1,2
3 1,2
4 2,5
5 5
6 --
3/ 0
M(T) rsum(a) M(T) 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).
00
1
234 334
2/ 2
1/ 0 4/ 1
3/ 2
0
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
Tutte polynomial of GG-parking functions
Spanning trees of G
BFSbijections
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
4
2
5
3
1
An example
0
4/
2/
5/
1/
t Queue
0 0
1 3,4
2 4,1,2
3 1,2
4 2,5
5 5
6 --
3/
f(v)={ (u,v) in E(G): rank(u)<rank(parent of v) }
0
4/ 0
2/ 1
5/ 2
1/ 0
t Queue
0 0
1 3,4
2 4,1,2
3 1,2
4 2,5
5 5
6 --
3/ 0
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
4/ 0
2/ 1
5/ 2
1/ 0
t Queue
0 03/ 0
Example
0
4/ -1
2/ 0
5/ 1
1/ 0
t Q
0 0
1 3,4
3/ -1
Example
0
4/ -2
2/ -1
5/ 1
1/ -1
t Q
0 0
1 3,4
2 4,1,2
3/-1
Example
0
4/ -2
2/ -2
5/ 0
1/ -2
t Q
0 0
1 3,4
2 4,1,2
3/-1
3 1,2
Example
0
4/ -2
2/ -3
5/ -1
1/ -2
t Q
0 0
1 3,4
2 4,1,2
3/-1
3 1,2
4 2,5
Example
0
4/ -2
2/ -3
5/ -2
1/ -2
t Q
0 0
1 3,4
2 4,1,2
3/-1
3 1,2
4 2,5
5 5
Example
0
4/ -2
2/ -3
5/ -2
1/ -2
t Q
0 0
1 3,4
2 4,1,2
3/-1
3 1,2
4 2,5
5 56 --
The ending value records the number of “extra edges”.
|E(G)|= v f(v) +|E(T)| +|Ex(T) in G|
Example
0
4/ -2
2/ -3
5/ -2
1/ -2
t Q
0 0
1 3,4
2 4,1,2
3/-1
3 1,2
4 2,5
5 56 --
Conclusion
Let rsum(f) = |E(G)|-n-v f(v)
One can get the full tG(x,y) by allowing multiroots for the
G-parking function.
Theorem.
tG(1,y) = f yrsum(f)
Theorem.
tG(1,y) = f yrsum(f)
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+1f (xy)r(f)-1y-sum(f)-Rec(f) ,
where Rec(f) is the number of edges incident to roots.
Theorem.
tG(1+x,y) = yE(G)-n+1f (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 left-to-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), (0,0)
TK3 = x2 + x + y.