graph homomorphisms between treesmath.sjtu.edu.cn/conference/bannai/2014/data/20141213a/...graph...

Graph homomorphisms between trees

Zhicong Lin

Jimei University

Joint work with Peter Csikvari (arXiv:1307.6721)

Zhicong Lin Graph homomorphisms between trees

http://arxiv-web3.library.cornell.edu/abs/1307.6721

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

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Figure : 6 endomorphisms of the path P3.


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

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The Path and the Star

Pn :=Path on n verticesSn :=Star on n vertices

Theorem (L. & Zeng, 2011)

|End(Pn)| =

{(n + 1)2n−1 − (2n − 1)

( n−1(n−1)/2

)if n is odd

(n + 1)2n−1 − n( nn/2

)if n is even

|End(Sn)| = (n − 1)n−1 + (n − 1)

How to compute |End(Tn)| for general trees Tn?

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HHH ss��s ��

�s

HHHs

ss

Figure : The path P6 and the star S7.


Main result

hom(Pn,Pn) ≤ hom(Pn,Tn) ≤ hom(Pn, Sn)

≥ ? ≥

hom(Tn,Pn)≤ hom(Tn,Tn) X hom(Tn, Sn)≥ ≥ ≥

hom(Sn,Pn) ≤ hom(Sn,Tn)≤ hom(Sn,Sn)

Figure : Trees with the same size

Theorem (L. & Csikvari)

For all trees Tn on n vertices we have

|End(Pn)| ≤ |End(Tn)| ≤ |End(Sn)|.


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 ) := (h1, h2, . . . , hm),

wherehi = |{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 )‖ =∑

hi .



An example:

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h(P3, v ,P3) = (1, 4, 1)



Lemma

Let G be a labeled graph and A = AG the adjacency matrix of G .Then the (i , j)-entry of the matrix An counts the number of walksin G from vertex i to vertex j with length n.

By this lemma, we have h(Pn, v ,G ) = 1An−1, where v is the initial(terminal) vertex of the path Pn and 1 denotes the row vector withall entries 1.We will generalize this to compute h(T , v ,G ).



@@@@ ss�

��

ss @@sT1

T2

s s s sv1 v

h(T1 ∪ T2, v1,G ) = h(T1, v1,G ) ∗ h(T2, v1,G )a ∗ b := (a1b1, . . . , anbn) is Hadamard product

h(T , v ,G ) = h(T1 ∪ T2, v1,G )A3


Sidorenko’s theorem on extremality of stars

Theorem (Sidorenko, 1994)

Let G be an arbitrary graph and let Tm be a tree on m vertices.Then

hom(Tm,G ) ≤ hom(Sm,G ).

Fiol & Garriga (2009) reproved the special case Tm = Pm.

Use Wiener index and some easy observations we (rediscoverand) give a new proof of Sidorenko’s theorem.

We constructe some special trees T to disprove the inequality

hom(Pm,T ) ≤ hom(Tm,T ).



≥ ? ≥

hom(Tn,Pn)≤ hom(Tn,Tn) X hom(Tn, Sn)

≥ ≥ ≥




KC-transformation

kk−1. . .

k−1

k

B A BA

x z

y

x

y

0 1

0

1

KC-transformation (Csikvari): A transformation on trees withrespect to the path 0, 1, . . . , k


KC-transformation

The KC-transformation give rise to a graded poset of trees on nvertices with the star as the largest and the path as the smallestelement.


KC-transformation: Closed Walks

Cn:=Cycle on n vertices

Theorem (Csikvari, 2010)

Let T be a tree and T ′ be a KC-transformation of T . Then

hom(Cm,T′) ≥ hom(Cm,T )

for any m ≥ 1.

The extremal problem about the number of closed walks in trees:

Corollary (Csikvari, 2010)

Let Tn be a tree on n vertices. We have

hom(Cm,Pn) ≤ hom(Cm,Tn) ≤ hom(Cm,Sn).


KC-transformation: Walks

Theorem (Bollobas & Tyomkyn, 2011)

Let T be a tree and T ′ be a KC-transformation of T . Then

hom(Pm,T′) ≥ hom(Pm,T )

for any m ≥ 1.

The extremal problem about the number of walks in trees:

Corollary (Bollobas & Tyomkyn, 2011)

Let Tn be a tree on n vertices. We have

hom(Pm,Pn) ≤ hom(Pm,Tn) ≤ hom(Pm,Sn).

A natural question arises: Does the above inequalities still truewhen replacing Pm by any tree?


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 respectto a path of length k . Then the inequality

hom(H,T ′) ≥ hom(H,T )

holds when

k is even and H is any tree

or k is odd and H is a starlike tree.

Corollary

Let H be a starlike tree and Tn be a tree on n vertices. Then

hom(H,Pn) ≤ hom(H,Tn) ≤ hom(H,Sn).


Counterexamples to the odd case

hom(Tn,Pn)≤ hom(Tn,Tn) X hom(Tn,Sn)

Counterexamples to the second inequality:

Figure : The doublestar S∗10

For k ≥ 5 we have hom(S∗2k , S∗2k)> hom(S∗2k , S2k). Note that S2k

can be obtained from S∗2k by a KC-transfromation.



≥ ? ≥


≥ ≥ ≥




A dual inequality

Bollobas & Tyomkyn’s theorem:

hom(Pn,Pm) ≤ hom(Pn,Tm) ≤ hom(Pn,Sm).

Theorem

Let Tm be a tree on m vertices and let T ′m be obtained from Tm

by a KC-transformation.

(i) If n is even, or n is odd and diam(Tm) ≤ n − 1, then

hom(Tm,Pn) ≤ hom(T ′m,Pn).

(ii) For any m, n,

hom(Pm,Pn) ≤ hom(Tm,Pn) ≤ hom(Sm,Pn).


A dual inequality

Figure : The trees T6 (left) and T ′6 (right).

The KC-transformation does not always increase the number ofhomomorphisms to the path Pn when n is odd. In the figure, wehave

hom(T6,P3) = 20 > 16 = hom(T ′6,P3).


A dual inequality

hom(Pm,Pn) ≤ hom(Tm,Pn) ≤ hom(Sm,Pn)

Our proof is complicated, which uses three treetransformations.


A dual inequality

T

T

T

T

TT

T

1 1

2 2

3 3

4

T T’

T4

Figure : LS-switch.


A dual inequality

T T’

Figure : Short-path shift (special KC-transformation).


A dual inequality

T’ T

Figure : Claw-deletion.


A dual inequality

Theorem

Let Tm be a tree on m vertices and let T ′m be obtained from Tm

by a KC-transformation.

(i) If n is even, or n is odd and diam(Tm) ≤ n − 1, then

hom(Tm,Pn) ≤ hom(T ′m,Pn).

(ii) For any m, n,

hom(Pm,Pn) ≤ hom(Tm,Pn) ≤ hom(Sm,Pn).

Idea of the proof:

(i) by the symmetry and unimodality of h(Tm,Pn).(ii) hom(Pm,Pn) ≤ hom(Tm,Pn)

n even: by (i).n odd: by the symmetry, bi-unimodal and log-concavity ofh(Tm,Pn) using the LS-switch, Short-path shift andClaw-deletion.



≥ ? ≥


≥ ≥ ≥




Markov chains and homomorphisms

Definition (Markov chains)

Let G be a graph with V (G ) = {1, 2, . . . , n}. Then P = (pij) is aMarkov chain on G if:∑

j∈N(i)

pij = 1 for all i ∈ V (G ),

where pij ≥ 0 and pij = 0 if (i , j) /∈ E (G ).

Definition (Stationary distribution)

Distribution Q = (qi ) is the stationary distribution of P if:∑j∈N(i)

qjpji = qi for all i ∈ V (G ).



Definition (Entropy)

We define the following three entropies:

H(Q) =∑

i∈V (G)

qi log1

qi,

andH(D|Q) =

∑i∈V (G)

qi log di ,

where di is the degree of i and let

H(P|Q) =∑

i∈V (G)

qi

( ∑j∈N(i)

pij log1

pij

).



Theorem

If Tm is a tree with ` leaves and m vertices, where m ≥ 3, then

hom(Tm,G ) ≥ exp

(H(Q) + `H(D|Q) + (m − 1− `)H(P|Q)

).

Corollary (Dellamonica et al., 2012)

Let G be a graph with e edges and degree sequence (d1, . . . , dn).Then for any treeTm with m vertices we have

hom(Tm,G ) ≥ 2e · Cm−2,

where C =(∏n

i=1 ddii

)1/2e.

Sketch of proof: Consider the classical Markov chain: pi ,j = 1di

ifj ∈ N(i).


Trees with 4 leaves

Lemma

If the tree Tn has at least four leaves, then

hom(Tm,Tn) ≥ (n − 2)2m−1 + 2.

Idea of the proof:

Fact: If G is a graph and G1,G2 are induced subgraphs of Gwith possible intersection, then for any graph H we have

hom(H,G ) ≥ hom(H,G1) + hom(H,G2)− hom(H,G1 ∩ G2).

We can reduce Tn to trees with exactly 4 leaves.

Use the LS-switch, we can further reduce Tn to 6 classes ofspecial trees with 4 leaves.

Construct some special Markov chains on the 6 classes oftrees and use the lower bound related to Markov chains.


Trees with 4 leaves

Theorem

If Tn is a tree on n vertices with at least 4 leaves, then

hom(Tm,Tn) ≥ hom(Tm,Pn).

Proof: Indeed,

hom(Tm,Tn)≥(n − 2)2m−1 + 2 = hom(Sm,Pn)≥ hom(Tm,Pn),

where the second inequality by Sidorenko’s theorem on extremalityof stars.


Summary of our results: trees with the same size


≥ ? ≥

hom(Tn,Pn)≤ hom(Tn,Tn) X hom(Tn, Sn)≥ ≥ ≥



Theorem (L. & Csikvari)

For all trees Tn on n vertices we have

|End(Pn)| ≤ |End(Tn)| ≤ |End(Sn)|.


Summary of our results: trees with different sizes

hom(Pm,Pn) ≤ hom(Pm,Tn) ≤ hom(Pm, Sn)

≥ X ≥

hom(Tm,Pn)(∗)≤ hom(Tm,Tn) X hom(Tm, Sn)

≥ ≥ ≥

hom(Sm,Pn) ≤ hom(Sm,Tn) ≤ hom(Sm,Sn)

Figure : Trees with different sizes.

The (∗) means that there are some well-determined (possible)counterexamples which should be excluded.


Further work

9 9

9

9

10 10 44

If we attach k copies of this rooted tree at the root then for theobtained tree Tm we have

hom(Tm,P4) = 2 · 4k + 2 · 10k > 4 · 9k = hom(Tm, S4)

for large enough k .

Conjecture

Let Tn be a tree on n vertices, where n ≥ 5. Then for any tree Tm

we havehom(Tm,Pn) ≤ hom(Tm,Tn).


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(K2,G )e(H) for all graph G .

Sidorenko’s conjecture is true for trees. Using our lower boundinvolving Markov chains we give a new proof of this fact.


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 vertexin A adjacent to all vertices in B (Conlon-Fox-Sudakov’10);

Cartesian product T�H, where T is a tree and H is abipartite graph satisfying Sidorenko’s conjecture(Kim-Lee-Lee’14);

Many other special bipartite graphs (by works of Blakley-Roy,Benjamini-Peres, Lovasz, Li-Szegedy and so on).

On the other hand, the simplest graph not known to haveSidorenko’s conjecture is K5,5 \ C10, a 3-regular graph on 10vertices.


Mobius function of a poset

Definition (Mobius function)

Let P be a poset. Define the Mobius function µ of P by

µ(s, s) = 1, for all s ∈ P

µ(s, u) = −∑

s≤t<u

µ(s, t), for all s < u in P.

Definition (Alternates in sign)

Let P be a grated poset with 0 and 1. We say that the Mobiusfunction of P alternates in sign if

(−1)`(s,t)µ(s, t) ≥ 0, for all s ≤ t in P.


Further work

Figure : EL-Shellable? Cohen-Maculay? Mobius functions on theintervals alternate in sign?


References

B. Bollobas and M. TyomkynWalks and paths in trees,J. Graph Theory, 70 (2012), 54-66.

P. CsikvariOn a poset of trees,Combinatorica, 30 (2010) 125-137.

Z. Lin and J. ZengOn the number of congruence classes of paths,Discrete Math., 312 (2012), 1300-1307.

A. SidorenkoA partially ordered set of functionals corresponding to graphs,Discrete Math., 131 (1994), 263-277.

Thanks!Zhicong Lin Graph homomorphisms between trees

http://onlinelibrary.wiley.com/doi/10.1002/jgt.20600/abstract

http://www.springerlink.com/content/30j081281222qk44/

http://www.sciencedirect.com/science/article/pii/S0012365X11005851

http://www.sciencedirect.com/science/article/pii/0012365X94903883

graph homomorphisms between treesmath.sjtu.edu.cn/conference/bannai/2014/data/20141213a/...graph...

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