A Faber-Krahn type inequality for regular trees

Transcription

1 A Faber-Krahn type inequality for regular trees Josef Leydold Institut für Statistik, WU Wien Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p0/36

2 The Faber-Krahn Theorem Among all bounded domains with fixed volume, a ball has the lowest first Dirichlet eigenvalue Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p1/36

3 Graph Laplacian Let weights Laplacian of : be a Graph with vertex set, edge set adjacency matrix diagonal matrix with Contrary to the classical Laplace-Beltrami operator on manifolds, the graph Laplacian is defined as a positive operator Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p2/36

4 Rayleigh Quotient The associate Rayleigh quotient on a real valued function on is the fraction Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p3/36

5 Geometric realization The geometric realization of is a metric space, consisting of the vertices arcs of length, glued between for every edge Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p4/36

6 A Continuous Graph Laplacian Define two measures on : number of vertices cumulated length of edges (ie the Lebesgue measure of ) Let denote the set of all continuous functions on are differentiable on, which Introduce operator on by the Rayleigh quotient Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p5/36

7 A Continuous Graph Laplacian The eigenfunctions of the first eigenvalue of edgewise linear functions are have the same eigenvalues eigenfunctions (The restrictions of the eigenfunctions of to graph Laplacian eigenvectors Friedman, 1993) are the Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p6/36

8 Graph with Boundary A graph with boundary is a graph interior vertices boundary vertices edges between interior vertices (interior edges) edges between boundary interior vertices (boundary edges) Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p7/36

9 Dirichlet Operator for Graphs Restrict Rayleigh quotient to those functions which vanish at all boundary vertices, ie The discrete version of this operator lives on interior vertices adjacency matrix of diagonal matrix with Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p8/36

10 -regular Trees with Boundary We look at trees with the following properties: all interior vertices have degree all boundary vertices have degree, all interior edges have length, all boundary edges have lengths there is at least one interior vertex,, Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p9/36

11 -regular Trees with Boundary We look at trees with the following properties: all interior vertices have degree all boundary vertices have degree, all interior edges have length, all boundary edges have lengths there is at least one interior vertex,, We get such a graph by cutting a region out of an infinite regular tree with edges of length We insert a (new) boundary vertex where we have cut an edge Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p9/36

12 -regular Trees with Boundary Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p10/36

13 The Faber-Krahn Property Let denote the lowest (Dirichlet) eigenvalue of We say a -regular tree with boundary Faber-Krahn property, if has the for all -regular trees with boundary with Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p11/36

14 Balls A ball is -regular tree with boundary with a center a radius,, such that dist for all boundary vertices dist denotes the geodesic distance between Amazingly, regular trees with the Faber-Krahn property are not balls Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p12/36

15 Onion-Shaped Tree Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p13/36

16 Branch Let dist ( root of tree) denote the height of the vertex For an edge the branch Br maximal subgraph induced by, of (ie the geodesic path at vertex is the all descendants contains ) Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p14/36

17 Branch The length Br is the maximal distance between boundary vertices The branch is called balanced if dist all boundary vertices Br is the same for Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p15/36

18 Onion-Shaped Tree // Definition We say a -regular tree with boundary onion-shaped if there exists a root that the following holds (O1) is connected is of the tree such Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p16/36

19 Onion-Shaped Tree // Definition We say a -regular tree with boundary onion-shaped if there exists a root that the following holds (O1) (O2) is connected then vertices ) Thus for an is of the tree such (if for all boundary Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p16/36

20 Onion-Shaped Tree // Definition We say a -regular tree with boundary onion-shaped if there exists a root that the following holds (O1) (O2) is connected then vertices ) Thus for an is of the tree such (if for all boundary (O3) All boundary edges have length or length, where is the same for all boundary edges of length Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p16/36

21 Onion-Shaped Tree // Definition We say a -regular tree with boundary onion-shaped if there exists a root that the following holds (O1) (O2) is connected then vertices ) Thus for an is of the tree such (if for all boundary (O3) All boundary edges have length or length, where is the same for all boundary edges of length (O4) If two branches Br Br, are not balanced, then Br Br, Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p16/36

22 A Faber-Krahn Theorem for Trees A -regular tree with boundary, Faber-Krahn property if only if the following holds:, has the is onion-shaped Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p17/36

23 A Faber-Krahn Theorem for Trees A -regular tree with boundary, Faber-Krahn property if only if the following holds:, has the is onion-shaped (F0) There is only one interior vertex, ie Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p17/36

24 A Faber-Krahn Theorem for Trees A -regular tree with boundary, Faber-Krahn property if only if the following holds:, has the is onion-shaped (F0) There is only one interior vertex, ie (F1) All branches of length are balanced, there is at most one balanced branch of length, provided that, or, or Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p17/36

25 The ball Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p18/36

26 A Faber-Krahn Theorem for Trees (cont) (F2) All branches of length are balanced, there is at most one balanced branch of length, provided that, or Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p19/36

27 A Faber-Krahn Theorem for Trees (cont) (F2) All branches of length are balanced, there is at most one balanced branch of length, provided that, or (F3) All branches of length are balanced, there is at most one balanced branch of length, provided that Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p19/36

28 A Faber-Krahn Theorem for Trees (cont) (F2) All branches of length are balanced, there is at most one balanced branch of length, provided that, or (F3) All branches of length are balanced, there is at most one balanced branch of length, provided that For a given volume homomorhpism, is uniquely defined up to Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p19/36

29 Example A 6-regular tree with the Faber-Krahn property:,, 4 interior vertices Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p20/36

30 Examples Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p21/36

31 Some basic properties of is a positive operator, ie An eigenfunction to the eigenvalue is either positive or negative on all interior vertices of is continuous as a function of is monotone in, ie if is a simple eigenvalue in the metric then For -regular trees with boundary we have Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p22/36

32 Spiral-like Ordering (Pruss 1998) We say that a well-ordering on is spiral-like providing the following conditions hold for all vertices : (S1) If (S2) If (S3) If then then is a child of (ie ), for, then are boundary edges of lengths, respectively, with, In (S3) the ordering of some boundary vertices is reverse to their lengths Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p23/36

33 Geometry of Eigenfunctions Let be a -regular tree with boundary with the Faber-Krahn property Then (M1) (M2) is connected; ; for all boundary vertices (M3) There exists a spiral-like ordering, for all vertices (M4) The normal derivative of length is the same such that at all boundary edges of Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p24/36

34 Rearrangements Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p25/36

35 Rearrangements Remove two edges of length, respectively, Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p25/36

36 Rearrangements Remove two edges of length, respectively, replace these by the respective edges of length Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p25/36

37 Rearrangements We get a new graph with same vertex set edge set which again is a -regular tree new Lemma: Furthermore, if then Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p26/36

38 Rearrangements We get a new graph with same vertex set edge set which again is a -regular tree new Lemma: (obvious) Furthermore, if then Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p26/36

39 Rearrangements Proof: (Idea) Have to show: Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p27/36

40 Rearrangements Proof: (Idea) Have to show: is eigenfuntion to Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p27/36

41 Rearrangements Proof: (Idea) Have to show: is lowest eigenvalue Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p27/36

42 Rearrangements Proof: (Idea) Have to show: Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p27/36

43 Rearrangements Now define a ordering on the interior vertices of, such that if Then rearrange the interior edges as described to make spiral-like on (stepwise, beginning with maximum ) Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p28/36

44 Derivative at Boundary Edges Normal derivative of, at the boundary edge, of length is The average normal derivative of given by boundary edges is Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p29/36

45 Perturbation of Boundary Edges Replace each of these where edges by edges of length, Then the normal derivative is the same for all these boundary edges If such an edge is longer than, then replace all the edges by edges of lengths where Make length as great as possible, ie (either) one edge or has Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p30/36

46 Perturbation of Boundary Edges Again, we get a new graph with same vertex set new edge set which again is a -regular tree Lemma: Equality holds if only if for all Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p31/36

47 Perturbation of Boundary Edges Again, we get a new graph with same vertex set new edge set which again is a -regular tree Lemma: (by definition of average derivative ) Equality holds if only if for all Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p31/36

48 Perturbation of Boundary Edges Again, we get a new graph with same vertex set new edge set which again is a -regular tree Lemma: (by definition of average derivative ) Equality holds if only if for all Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p31/36

49 Perturbation of Boundary Edges Again, we get a new graph with same vertex set new edge set which again is a -regular tree Lemma: (by definition of average derivative ) Equality holds if only if for all Alternatively, use perturbation theory for linear operators (matrices) Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p31/36

50 Resumé 1 Using the above lemmata a recursive rearrangement of results in a graph with decreased first Dirichlet eigenvalue (M2) for all boundary vertices (M3) There exists a spiral-like ordering, for all vertices (M4) The normal derivative of length is the same such that at all boundary edges of ; Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p32/36

51 Resumé 1 Using the above lemmata a recursive rearrangement of results in a graph with decreased first Dirichlet eigenvalue (M2) for all boundary vertices (M3) There exists a spiral-like ordering, for all vertices (M4) The normal derivative of length is the same such that at all boundary edges of ; For onion-shaped property we have to show that all almost all branches are balanced Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p32/36

52 Recursion Formula for Branches denote a geodesic path Let in with,, are balanced branches then, If Br is the length of the boundary edge where Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p33/36

53 Recursion Formula for Branches Consequently, we can express by where edge denotes the normal derivative at the boundary, ie The coefficients by the recursions are polynomials which are given, Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p34/36

54 Proof of the theorem By means of this recursion formula, the normal derivative at the leaves of balanced branches at the same vertex can be compared For the proof it is shown that many cases that are not excluded by Resumé 1 cannot exist (And it is too tedious to give any details),, respectively are the zeros of Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p35/36

55 References [1] J Friedman Some geometric aspects of graphs their eigenfunctions Duke Math J, 69(3), , 1993 [2] J Leydold A Faber-Krahn-Type Inequality for Regular Trees GAFA 7, , 1997 [3] A R Pruss Discrete convolution-rearrangement inequalities the Faber-Krahn inequality on regular trees Duke Math J, 91(3), , 1998 [4] J Leydold The geometry of regular trees with the Faber-Krahn property Discr Math 245, , 2002 Leydold 2003/02/13 A Faber-Krahn type inequality for regular trees p36/36