G-Parking Functions, Graph Searching, and Tutte Polynomial

1. Huafei Yan Nankai University andTexas A&M University Joint with Dimitrije Kostic G-Parking Functions, Graph Searching, and Tutte Polynomial

2. 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)

3. BFS on H 0 3 4 2 1 5

4. BFS on H 0 t Queue 3 4 0 0 2 1 5

5. BFS on H 0 t Queue 3 4 0 0 1 3,4 2 1 5

6. BFS on H 0 t Queue 3 4 0 0 1 3,4 2 4,1,2 2 1 5

7. BFS on H 0 t Queue 3 4 0 0 1 3,4 2 4,1,2 2 3 1,2 1 5

8. BFS on H 0 t Queue 3 4 0 0 1 3,4 2 4,1,2 2 3 1,2 1 4 2,5 5

9. BFS on H 0 t Queue 3 4 0 0 1 3,4 2 4,1,2 2 3 1,2 1 4 2,5 5 5 5 6 --

10. Which H s.t. BFS(H)=T? 0 t Queue 3 4 0 0 1 3,4 2 4,1,2 2 3 1,2 1 4 2,5 5 5 5 6 --

11. [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) ] .

12. 2. A familiar statistics Let M(T)=|Ex(T)|. Then number of labeled connected graphs on n+1 vertices with n+k edges is

13. 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).

14. Same property holds for external activity of trees inversion of trees level in recurrent configurations of sandpile model and (reversed) sum of parking functions…..

15. 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.

16. Reversed sum of PFs Let a=(a1,a2,…, an) be a PF. The reversed sumrsum(a) is i (i-1-ai) = n(n-1)/2-i ai

17. Reversed sum of PFs Let a=(a1,a2,…, an) be a PF. The reversed sumrsum(a) is i (i-1-ai) = n(n-1)/2-i ai rsum(a) has the same distribution as M(T).

18. 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|.

19. An example 0 t Queue 3/ 4/ 0 0 1 3,4 2 4,1,2 2/ 3 1,2 1/ 4 2,5 5 5 5/ 6 --

20. a(v) = rank of the parent of v 0 t Queue 3/ 0 4/ 0 0 0 1 3,4 2 4,1,2 2/ 1 3 1,2 1/ 1 4 2,5 5 5 5/ 3 6 --

21. a(v) = rank of the parent of v 0 t Q 3/ 0 4/ 0 0 0 1 3,4 2 4,1,2 2/ 1 3 1,2 1/ 1 4 2,5 5 5 5/ 3 6 -- M(T)rsum(a)

22. 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).

23. 0 0 0 4/ 1 1 1/ 0 2/ 2 3/ 2 3 234 34

24. 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.

25. General picture G-parking functions Tutte polynomial of G bijections BFS Spanning trees of G

26. 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.

27. 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) inG| where T ranges over all spanning trees of G.

28. 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.

29. An example 0 3 4 1 2 5

30. An example 0 t Queue 3/ 4/ 0 0 1 3,4 2 4,1,2 2/ 3 1,2 1/ 4 2,5 5 5 5/ 6 --

31. f(v)={ (u,v) in E(G): rank(u)<rank(parent of v) } 0 t Queue 3/0 4/ 0 0 0 1 3,4 2 4,1,2 2/ 1 3 1,2 1/ 0 4 2,5 5 5 5/ 2 6 --

32. 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.

33. Example 0 t Queue 3/0 4/ 0 0 0 2/ 1 1/ 0 5/ 2

34. Example 0 t Q 3/-1 4/ -1 0 0 1 3,4 2/ 0 1/ 0 5/ 1

35. Example 0 t Q 3/-1 4/ -2 0 0 1 3,4 2 4,1,2 2/ -1 1/ -1 5/ 1

36. Example 0 t Q 3/-1 4/ -2 0 0 1 3,4 2 4,1,2 2/ -2 1/ -2 3 1,2 5/ 0

37. Example 0 t Q 3/-1 4/ -2 0 0 1 3,4 2 4,1,2 2/ -3 1/ -2 3 1,2 4 2,5 5/ -1

38. Example 0 t Q 3/-1 4/ -2 0 0 1 3,4 2 4,1,2 2/ -3 1/ -2 3 1,2 4 2,5 5 5 5/ -2

39. Example 0 t Q 3/-1 4/ -2 0 0 1 3,4 2 4,1,2 2/ -3 1/ -2 3 1,2 4 2,5 5 5 5/ -2 6 --

40. The ending value records the number of “extra edges”. |E(G)|= v f(v) +|E(T)| +|Ex(T) in G|

41. Example 0 t Q 3/-1 4/ -2 0 0 1 3,4 2 4,1,2 2/ -3 1/ -2 3 1,2 4 2,5 5 5 5/ -2 6 --

42. 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)

43. 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).

44. 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.

45. 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}

46. 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.