Graph homomorphisms between trees

Transcription

1 Jimei University Joint work with Petér Csikvári (arxiv: )

2 Graph homomorphism Homomorphism: adjacency-preserving map f : V (G) V (H) uv E(G) = f (u)f (v) E(H) Hom(G, H) := the set of homomorphisms from G to H hom(g, H) := Hom(G, H)

3 Graph homomorphism Homomorphism: adjacency-preserving map f : V (G) V (H) uv E(G) = f (u)f (v) E(H) Hom(G, H) := the set of homomorphisms from G to H hom(g, H) := Hom(G, H) Endomorphism: a homomorphism from the graph to itself End(G) := the set of all endomorphisms of G Note: End(G) forms a monoid

4 Graph homomorphism Homomorphism: adjacency-preserving map f : V (G) V (H) uv E(G) = f (u)f (v) E(H) Hom(G, H) := the set of homomorphisms from G to H hom(g, H) := Hom(G, H) Endomorphism: a homomorphism from the graph to itself End(G) := the set of all endomorphisms of G Note: End(G) forms a monoid Figure : 6 endomorphisms of the path P 3.

5 The Path and the Star P n :=Path on n vertices S n :=Star on n vertices Theorem (L. & Zeng, 2011) { (n + 1)2 n 1 (2n 1) ( n 1 End(P n ) = (n + 1)2 n 1 n ( ) n n/2 (n 1)/2 ) if n is odd if n is even End(S n ) = (n 1) n 1 + (n 1) How to compute End(T n ) for general trees T n? Figure : The path P 6 and the star S 7.

6 Main result hom(p n, P n ) hom(p n, T n ) hom(p n, S n )? hom(t n, P n ) hom(t n, T n ) X hom(t n, S n ) hom(s n, P n ) hom(s n, T n ) hom(s n, S n ) Figure : Trees with the same size Theorem (L. & Csikvári) For all trees T n on n vertices we have End(P n ) End(T n ) End(S n ).

7 The Tree-walk algorithm How to count hom(t, G) for a tree T and a graph G?

8 The Tree-walk algorithm How to count hom(t, G) for a tree T and a graph G? Definition Let v V (T ) and V (G) = {1, 2,..., m}. Define h(t, v, G) := (h 1, h 2,..., h m ), where h i = {f Hom(T, G) f (v) = i}. We call h(t, v, G) the hom-vector at v from T to G. It is clear that hom(t, G) = h(t, v, G) = h i.

9 The Tree-walk algorithm An example: v 2 v 2 3 v v 1 v 2 v 2 Figure : 6 endomorphisms of the path P 3. h(p 3, v, P 3 ) = (1, 4, 1)

10 The Tree-walk algorithm Lemma Let G be a labeled graph and A = A G the adjacency matrix of G. Then the (i, j)-entry of the matrix A n counts the number of walks in G from vertex i to vertex j with length n. By this lemma, we have h(p n, v, G) = 1A n 1, where v is the initial (terminal) vertex of the path P n and 1 denotes the row vector with all entries 1. We will generalize this to compute h(t, v, G).

11 The Tree-walk algorithm T 1 v 1 v T 2 h(t 1 T 2, v 1, G) = h(t 1, v 1, G) h(t 2, v 1, G) a b := (a 1 b 1,..., a n b n ) is Hadamard product h(t, v, G) = h(t 1 T 2, v 1, G)A 3

12 Sidorenko s theorem on extremality of stars Theorem (Sidorenko, 1994) Let G be an arbitrary graph and let T m be a tree on m vertices. Then hom(t m, G) hom(s m, G).

13 Sidorenko s theorem on extremality of stars Theorem (Sidorenko, 1994) Let G be an arbitrary graph and let T m be a tree on m vertices. Then hom(t m, G) hom(s m, G). Fiol & Garriga (2009) reproved the special case T m = P m.

14 Sidorenko s theorem on extremality of stars Theorem (Sidorenko, 1994) Let G be an arbitrary graph and let T m be a tree on m vertices. Then hom(t m, G) hom(s m, G). Fiol & Garriga (2009) reproved the special case T m = P m. Use Wiener index and some easy observations we (rediscover and) give a new proof of Sidorenko s theorem.

15 Sidorenko s theorem on extremality of stars Theorem (Sidorenko, 1994) Let G be an arbitrary graph and let T m be a tree on m vertices. Then hom(t m, G) hom(s m, G). Fiol & Garriga (2009) reproved the special case T m = P m. Use Wiener index and some easy observations we (rediscover and) give a new proof of Sidorenko s theorem. We constructe some special trees T to disprove the inequality hom(p m, T ) hom(t m, T ).

16 hom(p n, P n ) hom(p n, T n ) hom(p n, S n )? hom(t n, P n ) hom(t n, T n ) X hom(t n, S n ) hom(s n, P n ) hom(s n, T n ) hom(s n, S n ) Figure : Trees with the same size

17 KC-transformation k 1 k x z y 0 x 1 k 1 y k A B A B KC-transformation (Csikvári): A transformation on trees with respect to the path 0, 1,..., k

18 KC-transformation The KC-transformation give rise to a graded poset of trees on n vertices with the star as the largest and the path as the smallest element.

19 KC-transformation: Closed Walks C n :=Cycle on n vertices Theorem (Csikvári, 2010) Let T be a tree and T be a KC-transformation of T. Then hom(c m, T ) hom(c m, T ) for any m 1. The extremal problem about the number of closed walks in trees: Corollary (Csikvári, 2010) Let T n be a tree on n vertices. We have hom(c m, P n ) hom(c m, T n ) hom(c m, S n ).

20 KC-transformation: Walks Theorem (Bollobás & Tyomkyn, 2011) Let T be a tree and T be a KC-transformation of T. Then for any m 1. hom(p m, T ) hom(p m, T ) The extremal problem about the number of walks in trees: Corollary (Bollobás & Tyomkyn, 2011) Let T n be a tree on n vertices. We have hom(p m, P n ) hom(p m, T n ) hom(p m, S n ). A natural question arises: Does the above inequalities still true when replacing P m by any tree?

21 Generalize to Tree-Walks Starlike tree: at most one vertex of degree greater than 2 Theorem Let T be a tree and T the KC-transformation of T with respect to a path of length k. Then the inequality holds when Corollary k is even and H is any tree hom(h, T ) hom(h, T ) or k is odd and H is a starlike tree. Let H be a starlike tree and T n be a tree on n vertices. Then hom(h, P n ) hom(h, T n ) hom(h, S n ).

22 Counterexamples to the odd case hom(t n, P n ) hom(t n, T n ) X hom(t n, S n ) Counterexamples to the second inequality: Figure : The doublestar S 10 For k 5 we have hom(s2k, S 2k )> hom(s 2k, S 2k). Note that S 2k can be obtained from S2k by a KC-transfromation.

23 hom(p n, P n ) hom(p n, T n ) hom(p n, S n )? hom(t n, P n ) hom(t n, T n ) X hom(t n, S n ) hom(s n, P n ) hom(s n, T n ) hom(s n, S n ) Figure : Trees with the same size

24 A dual inequality Bollobás & Tyomkyn s theorem: hom(p n, P m ) hom(p n, T m ) hom(p n, S m ). Theorem Let T m be a tree on m vertices and let T m be obtained from T m by a KC-transformation. (i) If n is even, or n is odd and diam(t m ) n 1, then (ii) For any m, n, hom(t m, P n ) hom(t m, P n ). hom(p m, P n ) hom(t m, P n ) hom(s m, P n ).

25 A dual inequality Figure : The trees T 6 (left) and T 6 (right). The KC-transformation does not always increase the number of homomorphisms to the path P n when n is odd. In the figure, we have hom(t 6, P 3 ) = 20 > 16 = hom(t 6, P 3 ).

26 A dual inequality hom(p m, P n ) hom(t m, P n ) hom(s m, P n ) Our proof is complicated, which uses three tree transformations.

27 A dual inequality T 1 1 T T 2 2 T T 4 T T 3 3 T T T 4 Figure : LS-switch.

28 A dual inequality T T Figure : Short-path shift (special KC-transformation).

29 A dual inequality T T Figure : Claw-deletion.

30 A dual inequality Theorem Let T m be a tree on m vertices and let T m be obtained from T m by a KC-transformation. (i) If n is even, or n is odd and diam(t m ) n 1, then (ii) For any m, n, hom(t m, P n ) hom(t m, P n ). hom(p m, P n ) hom(t m, P n ) hom(s m, P n ). Idea of the proof: (i) by the symmetry and unimodality of h(t m, P n ). (ii) hom(p m, P n ) hom(t m, P n ) n even: by (i). n odd: by the symmetry, bi-unimodal and log-concavity of h(t m, P n ) using the LS-switch, Short-path shift and Claw-deletion.

31 hom(p n, P n ) hom(p n, T n ) hom(p n, S n )? hom(t n, P n ) hom(t n, T n ) X hom(t n, S n ) hom(s n, P n ) hom(s n, T n ) hom(s n, S n ) Figure : Trees with the same size

32 Markov chains and homomorphisms Definition (Markov chains) Let G be a graph with V (G) = {1, 2,..., n}. Then P = (p ij ) is a Markov chain on G if: p ij = 1 for all i V (G), j N(i) where p ij 0 and p ij = 0 if (i, j) / E(G). Definition (Stationary distribution) Distribution Q = (q i ) is the stationary distribution of P if: q j p ji = q i for all i V (G). j N(i)

33 Markov chains and homomorphisms Definition (Entropy) We define the following three entropies: and H(Q) = H(D Q) = i V (G) where d i is the degree of i and let H(P Q) = i V (G) i V (G) q i ( q i log 1 q i, j N(i) q i log d i, p ij log 1 ). p ij

34 Markov chains and homomorphisms Theorem If T m is a tree with l leaves and m vertices, where m 3, then ( ) hom(t m, G) exp H(Q) + lh(d Q) + (m 1 l)h(p Q). Corollary (Dellamonica et al., 2012) Let G be a graph with e edges and degree sequence (d 1,..., d n ). Then for any treet m with m vertices we have ( n ) 1/2e. where C = i=1 d d i i hom(t m, G) 2e C m 2, Sketch of proof: Consider the classical Markov chain: p i,j = 1 d i j N(i). if

35 Trees with 4 leaves Lemma If the tree T n has at least four leaves, then Idea of the proof: hom(t m, T n ) (n 2)2 m Fact: If G is a graph and G 1, G 2 are induced subgraphs of G with possible intersection, then for any graph H we have hom(h, G) hom(h, G 1 ) + hom(h, G 2 ) hom(h, G 1 G 2 ). We can reduce T n to trees with exactly 4 leaves. Use the LS-switch, we can further reduce T n to 6 classes of special trees with 4 leaves. Construct some special Markov chains on the 6 classes of trees and use the lower bound related to Markov chains.

36 Trees with 4 leaves Theorem If T n is a tree on n vertices with at least 4 leaves, then Proof: Indeed, hom(t m, T n ) hom(t m, P n ). hom(t m, T n )(n 2)2 m = hom(s m, P n ) hom(t m, P n ), where the second inequality by Sidorenko s theorem on extremality of stars.

37 Summary of our results: trees with the same size hom(p n, P n ) hom(p n, T n ) hom(p n, S n )? hom(t n, P n ) hom(t n, T n ) X hom(t n, S n ) hom(s n, P n ) hom(s n, T n ) hom(s n, S n ) Figure : Trees with the same size Theorem (L. & Csikvári) For all trees T n on n vertices we have End(P n ) End(T n ) End(S n ).

38 Summary of our results: trees with different sizes hom(p m, P n ) hom(p m, T n ) hom(p m, S n ) X hom(t m, P n ) ( ) hom(t m, T n ) X hom(t m, S n ) hom(s m, P n ) hom(s m, T n ) hom(s m, S n ) Figure : Trees with different sizes. The ( ) means that there are some well-determined (possible) counterexamples which should be excluded.

39 Further work If we attach k copies of this rooted tree at the root then for the obtained tree T m we have hom(t m, P 4 ) = 2 4 k k > 4 9 k = hom(t m, S 4 ) for large enough k. Conjecture Let T n be a tree on n vertices, where n 5. Then for any tree T m we have hom(t m, P n ) hom(t m, T n ).

40 Sidorenko s conjecture Definition The homomorphism density t(h, G) is defined as follows: t(h, G) = hom(h, G). V (G) V (H) This is the probability that a random map is a homomorphism.

41 Sidorenko s conjecture Definition The homomorphism density t(h, G) is defined as follows: t(h, G) = hom(h, G). V (G) V (H) This is the probability that a random map is a homomorphism. Conjecture (Sidorenko, 1993) For every bipartite graph H with e(h) edges, t(h, G) t(k 2, G) e(h) for all graph G. Sidorenko s conjecture is true for trees. Using our lower bound involving Markov chains we give a new proof of this fact.

42 Sidorenko s conjecture Sidorenko s conjecture is true for Trees, even cycles and complete bipartite graph (Sidorenko);

43 Sidorenko s conjecture Sidorenko s conjecture is true for Trees, even cycles and complete bipartite graph (Sidorenko); Hypercubes (Hatami 10);

44 Sidorenko s conjecture Sidorenko s conjecture is true for Trees, even cycles and complete bipartite graph (Sidorenko); Hypercubes (Hatami 10); Bipartite graph H with bipartition A B and there is a vertex in A adjacent to all vertices in B (Conlon-Fox-Sudakov 10);

45 Sidorenko s conjecture Sidorenko s conjecture is true for Trees, even cycles and complete bipartite graph (Sidorenko); Hypercubes (Hatami 10); Bipartite graph H with bipartition A B and there is a vertex in A adjacent to all vertices in B (Conlon-Fox-Sudakov 10); Cartesian product T H, where T is a tree and H is a bipartite graph satisfying Sidorenko s conjecture (Kim-Lee-Lee 14);

46 Sidorenko s conjecture Sidorenko s conjecture is true for Trees, even cycles and complete bipartite graph (Sidorenko); Hypercubes (Hatami 10); Bipartite graph H with bipartition A B and there is a vertex in A adjacent to all vertices in B (Conlon-Fox-Sudakov 10); Cartesian product T H, where T is a tree and H is a bipartite graph satisfying Sidorenko s conjecture (Kim-Lee-Lee 14); Many other special bipartite graphs (by works of Blakley-Roy, Benjamini-Peres, Lovász, Li-Szegedy and so on).

47 Sidorenko s conjecture Sidorenko s conjecture is true for Trees, even cycles and complete bipartite graph (Sidorenko); Hypercubes (Hatami 10); Bipartite graph H with bipartition A B and there is a vertex in A adjacent to all vertices in B (Conlon-Fox-Sudakov 10); Cartesian product T H, where T is a tree and H is a bipartite graph satisfying Sidorenko s conjecture (Kim-Lee-Lee 14); Many other special bipartite graphs (by works of Blakley-Roy, Benjamini-Peres, Lovász, Li-Szegedy and so on). On the other hand, the simplest graph not known to have Sidorenko s conjecture is K 5,5 \ C 10, a 3-regular graph on 10 vertices.

48 Möbius function of a poset Definition (Möbius function) Let P be a poset. Define the Möbius function µ of P by µ(s, s) = 1, for all s P µ(s, u) = µ(s, t), for all s < u in P. s t<u Definition (Alternates in sign) Let P be a grated poset with ˆ0 and ˆ1. We say that the Möbius function of P alternates in sign if ( 1) l(s,t) µ(s, t) 0, for all s t in P.

49 Further work Figure : EL-Shellable? Cohen-Maculay? Möbius functions on the intervals alternate in sign?

50 References B. Bollobás and M. Tyomkyn Walks and paths in trees, J. Graph Theory, 70 (2012), P. Csikvári On a poset of trees, Combinatorica, 30 (2010) Z. Lin and J. Zeng On the number of congruence classes of paths, Discrete Math., 312 (2012), A. Sidorenko A partially ordered set of functionals corresponding to graphs, Discrete Math., 131 (1994), Thanks!