Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
Graph homomorphisms between trees
Zhicong Lin
Jimei University
Joint work with Peter Csikvari (arXiv:1307.6721)
Zhicong Lin Graph homomorphisms between trees
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
sss
sss
-
-
-
sss
sss
-
-
AAAAAU s
ss
sss����@@@R
@@@R
sss
sss
��������@@@R
sss
sss
������
-
AAAAAU s
ss
sss
������
-
-
Figure : 6 endomorphisms of the path P3.
Zhicong Lin Graph homomorphisms between trees
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
sss
sss
-
-
-
sss
sss
-
-
AAAAAU s
ss
sss����@@@R
@@@R
sss
sss
��������@@@R
sss
sss
������
-
AAAAAU s
ss
sss
������
-
-
Figure : 6 endomorphisms of the path P3.
Zhicong Lin Graph homomorphisms between trees
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
sss
sss
-
-
-
sss
sss
-
-
AAAAAU s
ss
sss����@@@R
@@@R
sss
sss
��������@@@R
sss
sss
������
-
AAAAAU s
ss
sss
������
-
-
Figure : 6 endomorphisms of the path P3.
Zhicong Lin Graph homomorphisms between trees
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?
s��s@@s��s
@@s��s
HHH ss���s ��
�s
HHHs
ss
Figure : The path P6 and the star S7.
Zhicong Lin Graph homomorphisms between trees
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)|.
Zhicong Lin Graph homomorphisms between trees
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 .
Zhicong Lin Graph homomorphisms between trees
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 .
Zhicong Lin Graph homomorphisms between trees
The Tree-walk algorithm
An example:
sss
sss
-
-
-
v 2
sss
sss
-
-
AAAAAU
v 2
sss
sss����@@@R
@@@Rv
1 sss
sss
��������@@@Rv
3
sss
sss
������
-
AAAAAU
v 2
sss
sss
������
-
-
v 2
Figure : 6 endomorphisms of the path P3.
h(P3, v ,P3) = (1, 4, 1)
Zhicong Lin Graph homomorphisms between trees
The Tree-walk algorithm
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 ).
Zhicong Lin Graph homomorphisms between trees
The Tree-walk algorithm
@@@@ 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
Zhicong Lin Graph homomorphisms between trees
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 ).
Zhicong Lin Graph homomorphisms between trees
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 ).
Zhicong Lin Graph homomorphisms between trees
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 ).
Zhicong Lin Graph homomorphisms between trees
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 ).
Zhicong Lin Graph homomorphisms between trees
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
Zhicong Lin Graph homomorphisms between trees
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
Zhicong Lin Graph homomorphisms between trees
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.
Zhicong Lin Graph homomorphisms between trees
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).
Zhicong Lin Graph homomorphisms between trees
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?
Zhicong Lin Graph homomorphisms between trees
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).
Zhicong Lin Graph homomorphisms between trees
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.
Zhicong Lin Graph homomorphisms between trees
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
Zhicong Lin Graph homomorphisms between trees
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).
Zhicong Lin Graph homomorphisms between trees
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).
Zhicong Lin Graph homomorphisms between trees
A dual inequality
hom(Pm,Pn) ≤ hom(Tm,Pn) ≤ hom(Sm,Pn)
Our proof is complicated, which uses three treetransformations.
Zhicong Lin Graph homomorphisms between trees
A dual inequality
T
T
T
T
TT
T
1 1
2 2
3 3
4
T T’
T4
Figure : LS-switch.
Zhicong Lin Graph homomorphisms between trees
A dual inequality
T T’
Figure : Short-path shift (special KC-transformation).
Zhicong Lin Graph homomorphisms between trees
A dual inequality
T’ T
Figure : Claw-deletion.
Zhicong Lin Graph homomorphisms between trees
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.
Zhicong Lin Graph homomorphisms between trees
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
Zhicong Lin Graph homomorphisms between trees
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 ).
Zhicong Lin Graph homomorphisms between trees
Markov chains and homomorphisms
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
).
Zhicong Lin Graph homomorphisms between trees
Markov chains and homomorphisms
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).
Zhicong Lin Graph homomorphisms between trees
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.
Zhicong Lin Graph homomorphisms between trees
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.
Zhicong Lin Graph homomorphisms between trees
Summary of our results: trees with the same size
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)|.
Zhicong Lin Graph homomorphisms between trees
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.
Zhicong Lin Graph homomorphisms between trees
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).
Zhicong Lin Graph homomorphisms between trees
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.
Zhicong Lin Graph homomorphisms between trees
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.
Zhicong Lin Graph homomorphisms between trees
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.
Zhicong Lin Graph homomorphisms between trees
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.
Zhicong Lin Graph homomorphisms between trees
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.
Zhicong Lin Graph homomorphisms between trees
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.
Zhicong Lin Graph homomorphisms between trees
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.
Zhicong Lin Graph homomorphisms between trees
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.
Zhicong Lin Graph homomorphisms between trees
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.
Zhicong Lin Graph homomorphisms between trees
Further work
Figure : EL-Shellable? Cohen-Maculay? Mobius functions on theintervals alternate in sign?
Zhicong Lin Graph homomorphisms between trees
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