Homologies and cohomologies of digraphs Yong Lin Renmin University - - PowerPoint PPT Presentation

Homologies and cohomologies of digraphs

Yong Lin

Renmin University of China

July 8, 2014

This is a joined work with: Alexander Grigor’yan, Yuri Muranov, Shing-Tung Yau

1

Definition of homologies and cohomologies of digraphs

2

A differential calculus on an associative algebra A is an algebraic analogue of the calculus of differential forms on a smooth manifold. The discrete differential calculus is based on the universal differential calculus on an associative algebra of functions on discrete set. By a natural way, we can consider this calculus as a calculus on a universal digraph(complete digraph) with the given discrete set of vertices. This approach gives an opportunity to define differential calculus for every subgraph of the universal digraph.

3

Given a finite set V, we define a p-form ω on V as K-valued function on V p+1. The set of all p-forms is a linear space over K that is denoted by Λp. It has a canonical basis ei0...ip. For any ω ∈ Λp, we have ω =

• i0,...,ip∈V

ωi0...ipei0...ıp where ωi0...ip = ω(i0, . . . , ip). The exterior derivative d: Λp → Λp+1 is defined by (dω)i0...ip+1 =

p+1

• q=0

(−1)qωi0...ˆ

iq...ip+1

and satisfies d2 = 0.

4

An elementary p-path is any (ordered) sequence i0, ..., ip of p + 1 vertices of V that will be denoted simply by i0...ip or by ei0...ip. We use the notation ei0...ip when we consider the elementary path as an element of a linear space Λp = Λp (V) that consists of all formal linear combinations of all elementary p-paths. The elements of Λp are called p-paths. For any v ∈ Λp we have v =

• i0,...,ip∈V

vi0...ipei0...ip and a pairing with a p-path ω (ω, v) =

• i0,...,ip

ωi0...ipvi0...ip.

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The dual operator ∂ : Λp+1 → Λp is given by ∂ei0...ip+1 =

p+1

• q=0

(−1)qei0...ˆ

iq...ip+1

6

We say that a path i0...ip is regular if ik = ik+1 for all k = 0, ..., p − 1, and irregular otherwise. Consider the following subspace of Λp: Rp = span

• ei0...ip : i0...ip is regular
• =
• ω ∈ Λp : ωi0...ip = 0 if i0...ip is irregular
• ,

where span A denotes the linear space spanned by the set A. The elements of Rp are called regular p-forms. For example, ω ∈ R1 then ωii ≡ 0 and ω ∈ R2 then ωiij ≡ ωjii ≡ 0.

7

We say that an elementary p-path ei0...ip is regular (or irregular) if the path i0...ip is regular (resp. irregular). We would like to define the boundary operator ∂ on the subspace of Λp spanned by regular elementary paths. Just restriction of ∂ to the subspace does not work as ∂ is not invariant on this subspace, so that we have to consider a quotient space instead.

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Let Ip be the subspace of Λp that is spanned by irregular ei0...ip. Consider the quotient spaces Rp := Λp/Ip. The elements of Rp are the equivalence classes v mod Ip where v ∈ Λp, and they are called regularized p-paths.

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Lemma

Let p ≥ −1. If v1, v2 ∈ Λp and v1 = v2 mod Ip, then ∂v1 = ∂v2 mod Ip−1.

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Proof.

If p ≤ 0 there is nothing to prove since Ip = {0}. In the case p ≥ 1, it suffices to prove that if v = 0 mod Ip then ∂v = 0 mod Ip−1. It suffices to prove that if ei0...ip is non-regular then ∂ei0...ip is non-regular, too. Indeed, for a non-regular path i0...ip there exists an index k such that ik = ik+1. Then we have ∂ei0...ip = ei1...ip − ei0i2...ip + ... + (−1)k ei0...ik−1ik+1ik+2...ip + (−1)k+1 ei0...ik−1ikik+2...ip +... + (−1)p ei0...ip−1. By ik = ik+1 the two terms in the above line cancel out, whereas all other terms are non-regular, whence ∂ei0...ip ∈ Ip−1.

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Let i0 . . . ip be an elementary regular p-path on V. It is called allowed if ikik+1 ∈ E for any k = 0, . . . , p − 1, and non-allowed

• therwise. The set of all allowed elementary p-paths will be

denoted by Ep,and non-allowed by Np. For example E0 = V and E1 = E. Denote by Ap = Ap(V, E) the subspace of Rp spanned by the allowed elementary p-paths, that is, Ap = span{ei0...ip : i0 . . . ip ∈ Ep} = {v ∈ Rp : vi0...ip = 0∀i0 . . . ip ∈ Np} The elements of Ap are called allowed p-paths.

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Similarly, denote by N p the subspace of Rp, spanned by the non-allowed elementary p-forms, that is, N p = span{ei0...ip : i0 . . . ip ∈ Np} = {ω ∈ Rp : ωi0...ip = 0∀i0 . . . ip ∈ Ep} Clearly, we have Ap = (N p)⊥ where ⊥ refers to the annihilator subspace with respect to the couple (Rp, Rp) of dual spaces.

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We would like to reduce the space Rp of regular p-forms so that the non-allowed forms can be treated as zeros. Consider the following subspaces of spaces Rp J p ≡ J p(V, E) := N p + dN p−1 that are d-invariant, and define the space Ωp of p-forms on the digraph (V, E) by Ωp ≡ Ωp(V, E) := Rp/J p Then d is well-defined on Ωp and we obtain a cochain complex 0 → Ω0

d

− → . . .

d

− → Ωp

d

− → Ωp+1

d

− → . . . Shortly we write Ω• = R•/J • where Ω is the complex and R and J refer to the corresponding cochain complexes. If the digraph (V, E) is complete, that is, E = V × V \ diag then the spaces N p and J p are trivial, and Ωp = Rp.

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Consider the following subspaces of Ap Ωp ≡ Ωp(V, E) = {v ∈ Ap : ∂v ∈ Ap−1} that are ∂-invariant. Indeed, v ∈ Ωp ⇒ ∂v ∈ Ap−1 ⊂ Ωp−1. The elements of Ωp are called ∂-invariant p-paths. We obtain a chain complex Ω• 0 ← − Ω0

∂

← − Ω1

∂

← − . . .

∂

← − Ωp−1

∂

← − Ωp

∂

← − . . . that,in fact, is dual to Ω•.

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By construction we have Ω0 = A0 and Ω1 = A1 so that dimΩ0 =| V | and dimΩ1 =| E |, while in general Ωp ⊂ Ap. Let us define the cohomologies and homologies of the digraph (V, E) by Hp (V, E) := Hp (Ω) = ker d|Ωp /Im d|Ωp−1 . and Hp (V, E) := Hp (Ω) = ker ∂|Ωp

• Im ∂|Ωp+1.

Recall that Hp(V, E) and Hp(V, E) are dual and hence their dimensions are the same. The values of dimHp(V, E) can be regarded as invariants of the digraph (V, E). Note that for any p ≥ 0 dimHp(Ω) = dimΩp − dim∂Ωp − dim∂Ωp+1

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Sometimes it is useful to be able to determine the homology groups Hn directly via the spaces An, without Ωn, as in the next statement.

Proposition

We have Hn = ker ∂|An/ (An ∩ ∂An+1) (1) and dim Hn = dim An − dim ∂An − dim (An ∩ ∂An+1) . (2)

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Let us define the Euler characteristic of the digraph (V, E) by χ(V, E) =

n

• p=0

(−1)pdimHp(Ω) provided n is so big that dimHp(Ω) = 0 for all p > n.

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Example

Consider the graph of 6 vertices V = {0, 1, 2, 3, 5} with 8 edges E = {01, 02, 13, 14, 23, 24, 53, 54}.

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Let us compute the spaces Ωp and the homologies Hp(Ω). We have Ω0 = A0 = span{e0, e1, e2, e3, e4, e5}, dimΩ0 = 6 Ω1 = A1 = span{e01, e02, e13, e14, e23, e24, e53, e54}, dimΩ1 = 8 A2 = span{e013, e014, e023, e024}, dimA2 = 4 The set of semi-edges is S = {e03, e04} so that dimΩ2 = dimA2− | S |= 2. The basis in Ω2 can be easily spotted as each of two squares 0, 1, 2, 3 and 0, 1, 2, 4 determine a ∂-invariant 2-path, whence Ω2 = span{e013 − e023, e014 − e024} Since there are no allowed 3-paths, we see that A3 = Ω3 = {0}. It follows that χ = dimΩ0 − dimΩ1 + dimΩ2 = 6 − 8 + 2 = 0

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Let us compute dim H2: dimH2 = dimΩ2 − dim∂Ω2 − dim∂Ω3 = 2 − dim∂Ω2. The image ∂Ω2 is spanned by two 1-paths ∂(e013 − e023) = e13 − e03 + e01 − (e23 − e03 + e02) = e13 + e01 − e23 − e02 ∂(e014 − e024) = e14 − e04 + e01 − (e24 − e04 + e02) = e14 + e01 − e24 − e02 that are clearly linearly independent. Hence, dim∂Ω2 = 2 whence dimH2 = 0.

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The dimension of H1 can be computed similarly, but we can do easier using the Euler characteristic: dimH0 − dimH1 + dimH2 = χ = 0 whence dim H1 = 1. In fact, a non-trivial element of H1 is determined by 1-path v = e13 − e14 − e53 + e54 Indeed, by a direct computation ∂v = 0, so that v ∈ ker∂ |Ω1 while for v to be in Im∂ |Ω2, it should be a linear combination of ∂(e013 − e023) and ∂(e014 − e024), which is not possible since they do not have the term e54.

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Example

4 5 1 2 3

Figure: planar graph with non-trivial homology group H2

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A direct computation shows that H1 (G, K) = {0} and H2 (G, K) ∼ = K, where H2 (G) is generated by e124 + e234 + e314 − (e125 + e235 + e315) . It is easy to see that G is a planar graph but nevertheless its second homology group is non-zero. This shows that the digraph homologies “see” some non-trivial intrinsic dimensions

• f digraphs that are not necessarily related to embedding

properties.

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Let us call by a triangle a sequence of three distinct vertices a, b, c ∈ V such that ab, bc, ac are edges:

b

• a•

ր

→

ց•c

Let us called by a square a sequence of four distinct vertices a, b, b′, c ∈ V such that ab, bc, ab′, b′c are edges:

b

• a•ր

ց ց ր•c

• b′

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For a cycle-graph we have dim H0 = 1 and dim Ω0 = |V| = |E| = dim Ω1. (3)

Theorem

Let (V, E) be a cycle-graph. Then dim Ωp = 0 for all p ≥ 3 dim Hp (Ω) = 0 for all p ≥ 2. If (V, E) is a triangle or a square then dim Ω2 = 1, dim H1 (Ω) = 0, χ = 1 whereas otherwise dim Ω2 = 0, dim H1 (Ω) = 1, χ = 0.

26

We say that a digraph (V, E) is star-like if there is a vertex a ∈ V (called a star center) such that the edge (a, i) ∈ E for all i = a. For example, a digraph

ր

1

• ց

0• ⇆ •2 is star-like with the

star center 0. Of course, any complete digraph is star-like.

Theorem

If (V, E) is a star-like digraph, then dim Hp (V, E) = 0 for any p ≥ 1 and dim H0 (V, E) = 1.

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Theorem

Let a digraph (V, E) be a tree (that is, the underlying undirected graph is a tree). Then Hp (V, E) = 0 for all p ≥ 1.

Definition

Let G be a simple graph with the set of vertices V and the set

• f edges EG. Define a simple graph G with the same set of

vertices V and with the set of inverse-directed edges.

Theorem

Let G be a simple graph. Then we have an isomorphism of cochain complexes ΩG − → ΩG which is given on the basic elements by the following map ei0i1...ip−1ip − → (−1)keipip−1...i1i0, where k = 1 for p = 1, 2 mod 4 and k = 0 for p = 0, 3 mod 4.

28

Definition

The cone CG of the graph G is obtained by adding a new vertex v to the set of vertexes V and new edges {i, v} for all 0 ≤ i ≤ n. 3 4 2 1

Figure: Cone, v = 4.

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Definition

The suspension SG of the graph G is obtained from the graph G by adding two new vertices v and w and new edges {i, v}, {i, w} for all 0 ≤ i ≤ n. a b

Figure: Suspension, v = a, w = b.

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Theorem

For any graph G we have Hp(ΩCG) ∼ = K, p = 0 0, p ≥ 1, , Hp+1(ΩSG) ∼ =    K, p = −1 H0(Ωε

G),

p = 0 Hp(ΩG), p ≥ 1 where Ωε

G is a cochain complex with the augmentation.

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Let G1, G2 be graphs with pointed vertices v1, v2, respectively. Assume that all the vertex sets Vi are disjoint. The wedge sum (or bouquet) (G, v) of these graphs is a graph with the set V of vertices that is obtained from the disjoint union U = V1 V2 by identification of all pointed vertices v1 and v2 with the vertex v, and with the set of edges E = E1 E2 with the same identification of the endpoint. We shall denote the wedge sum by G = G1

• G2.

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Theorem

Let G = G1 G2 where Gi (i = 1, 2) are connected graphs. Then Hp(ΩG) =

• K,

p = 0 Hp(ΩG1) ⊕ Hp(ΩG2), p ≥ 1.

33

Theorem

For any finite collection of nonnegative integers k0, k1, . . . kn such that k0 ≥ 1 there exists a digraph G such that the cohomology groups of its differential calculus satisfies the conditions dim Hi(ΩG) = ki, for all 0 ≤ i ≤ n.

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Given two digraphs X and Y, define there Cartesian product digraph Z = X × Y as follows:

◮ the vertices of Z are the couples (a, b) where a ∈ VX and

b ∈ VY;

◮ the edges in Z are of two types: (a, b) → (a′, b) where

a → a′ (a horizontal edge) and (a, b) → (a, b′) where b → b′ (a vertical edge): b′• . . .

(a,b′)

• −

→

(a′,b′)

• . . .

↑ ↑ ↑ b• . . .

(a,b)

• −

→

(a′,b)

• . . .

Y X

. . .

• a

− →

• a′

. . .

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Theorem

Let P (X) and P (Y) be two regular path complexes. Then for their Cartesian product P (Z) = P (X) ⊞ P (Y) the following isomorphism of chain complexes holds: Ω• (Z) ∼ = Ω• (X) ⊗ Ω• (Y)

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Consequently, we obtain a K¨ unneth formula H• (Z) ∼ = H• (X) ⊗ H• (Y) , that is, for any r ≥ 0, Hr (Z) ∼ =

• {p,q≥0:p+q=r}

(Hp (X) ⊗ Hq (Y)) .

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Definition

For any digraph X, the cylinder over X is the digraph Cyl X := X ×

• 0• → •1

.

Theorem

There is an isomorphism Hp (Cyl X) = Hp (X) for all p ≥ 0.

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Example

Consider a (undirected) graph G with 6 vertices and 12 edges

• n the picture

3 4 1 2 3 5 4 2 3 1 5

Figure: G embedded on the Mobius band (left) and in real space

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As an one-dimensional simplicial complex, G has simplicial cohomologies H∗ (C∗ (G)). On the other hand, let us introduce arbitrarily a set D of directions on the edges of G, so that (G, D) is a digraph and, hence, has the graph cohomologies H∗ (G, D).

Theorem

For any choice of D, H1 (C∗ (G)) = H1 (G, D) . (4) More presice, dim H1 (C∗ (G)) = 7 and dim H1 (G, D) ≤ 6.

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For any finite simplicial complex S define a finite digraph GS. The set of vertices of GS coincides with the set of all simplexes from S, and two simplexes s, t are connected in GS by a directed edge s → t if and only if s ⊃ t and dim s = dim t + 1. (5)

(a) simplicial complex S (b) digraph GS based on BS (d) cubical complex QS (c) abstract digraph GS

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The graph GS can be realized geometrically as follows. Denote by bs the barycenter of a simplex s ∈ S, and consider the set BS of the barycenters of all s ∈ S. Define the edges bs → bt between two barycenters by the same rule (4), which makes GS into a digraph as in the picture. All homologies are considered over a fixed field K.

Theorem

For any finite simplicial complex S and for any n ≥ 0, we have isomorphism Hn (C∗ (S)) ∼ = Hn (GS) .

42

For any finite simplicial complex S define a finite digraph BS. The set of vertices of GS coincides with the set of all simplexes from S, and two simplexes s, t are connected in BS by a directed edge s → t if and only if s ⊃ t and s = t. (6)

Theorem

For any finite simplicial complex S and for any n ≥ 0, we have isomorphism Hn (C∗ (S)) ∼ = Hn (BS) .

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(a) simplicial complex S (b) digraph G based on B(S)

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Minimal paths and hole detection

45

The elements of Hp(V, E) can be regarded as p-dimensional holes in the digraph (V, E). To make this notion more geometric, we can work with representatives of the homologies classes, which are closed p-paths. We say that two closed p-paths u and v are homological and write u ∼ v if u and v represent the same homology class, that is, if u − v is exact. For any p-path v define its length by l(v) =

• i0,...,ip∈V

| vi0...ip |

46

Given a closed p-path v0, consider the minimization problem l(v) → min for v ∼ v0 This problem always has a solution, although not necessarily

• unique. Any solution is called a minimal p-path. It is hoped that

minimal p-paths(in a given homology class) match our geometric intuition of what holes in a graph should be. In this section we give some examples of minimal paths to support this claim.

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Example

Figure: Fig.1

Consider a digraph on Fig.1.

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Removing successively the vertices A, B, 8, 9, 6, 7 we obtain a digraph (V ′, E′) with V ′ = {0, 1, 2, 3, 4, 5} and E′ = {02, 03, 04, 05, 12, 13, 14, 15, 24, 25, 34, 35} that has the same homologies as (V, E). The digraph (V ′, E′) is shown in two ways on the following figure. Clearly the second representation of this graph is reminiscent of an octahedron.

Figure: Fig.2

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The digraph (V ′, E′) is the same as the 2-dimensional sphere-graph. Hence we obtain that dimH2(V, E) = 1 while Hp(V, E) = 0 for p = 1 and p > 2. Consider a closed 2-path on (V, E) v0 = e024 − e025 − e034 + e035 − e124 + e125 + e134 − e135 − e634 + e614 − e613 Then a solution to the minimization problem is given by v = e024 − e025 − e034 + e035 − e124 + e125 + e134 − e135 that is a 2-path that determines a 2-dimensional hole in (V, E) given by the octahedron. Note that on figure 1 this octahedron is hardly visible, but it can be computed purely algebraically as shown above.

50

Homotopy theory of graphs

51

we can also define a homotopy theory of digraphs and prove homotopy invariance of homologies theories of digraphs.

52

Definition

Let G = (VG, EG), H = (VH, EH) be (undirected) graphs. A morphism f : G → H is given by a map of vertices f : VG → VH such that if (v, w) ∈ EG then we have (f(v), f(w)) ∈ EH or f(v) = f(w) ∈ W. The last condition means that the edge (v, w) maps to the vertex f(v) = f(w).

53

Definition

The differential calculus on a graph G is the differential calculus ΩS(G) on the symmetric digraph S(G). The cohomology ring H∗(G, K) of a graph G is the cohomology ring H∗(S(G), K).

54

Recall a homotopy theory of graphs constructed by Barcelo et al. Let In(n ≥ 0) denote a graph with the set of vertices VI = {0, 1, . . . , n} and the set of edges either (i, i + 1) or (i + 1, i) for 0 ≤ i ≤ n − 1. The graph In we shall call a line

• graph. Note that by this definition a one-point graph I0 is a line
• graph. Let I = I1.

55

For two graphs G = (VG, EG) and H = (VH, EH) we define a ⊡- product Π = G ⊡ H = (VΠ, EΠ) as a graph with a set of vertices VΠ = VG × VH and a set of edges EΠ such that [(x, y), (x′, y′)] ∈ EΠ for x, x′ ∈ VG; y, y′ ∈ VH if one of the conditions is satisfied x′ = x, (y, y′) ∈ EH

• r y′ = y, (x, x′) ∈ EG.

56

Definition

i) Two maps fi : G → H, i = 0, 1

• f a graph G to a graph H are homotopic if there exists a line

graph In and a morphism F : G ⊡ In → H such that F|G⊡{0} = f0 : G ⊡ {0} → H, F|G⊡{n} = f1 : G ⊡ {n} → H. In this case we shall write f0 ≃ f1.

57

ii) Two graphs G and H are homotopy equivalent if there exist maps f : G → H, g : H → G such that f ◦ g ≃ IdH, g ◦ f ≃ IdG . In this case we shall write H ≃ G. In this case maps f and g are called homotopy inverses of each other.

58

The above defination of homotopy can be defined similar in digraphs. In the case n = 1 we refer to the map F as an one-step homotopy between f0 and f1. In this case the identities determine F uniquely, and the requirement is that the so defined F is a digraph map of G ⊡ I1 to H. Since for I1 there are

• nly two choices 0 → 1 and 0 ← 1, we obtain that f0 and f1 are
• ne-step homotopic, if

either f0 (x) − → =f1 (x) for all x ∈ VG

• r f1 (x) −

→ =f0 (x) for all x ∈ VG.

59

It follows that f0 and f1 are homotopic if there is a finite sequence of digraph maps f0 = g0, g1, ..., gn = f1 from G to H such that gk and gk+1 are one-step homotopic. It is obvious that the relation ”≃” is an equivalence relation on the set of all digraph maps from G to H.

60

Now we state the theorem that answer on the question from Barcelo et al about construction of natural homotopy invariant homology theory for graphs.

Theorem

i) Let f ≃ g : G → H be two homotopy equivalent maps of

• graphs. Then these maps induce the equal homomorphisms of

cohomology groups and homology groups. ii) Let G ≃ H be two homotopy equivalent graphs. Then they have isomorphic cohomology groups and homology groups.

61

Thanks !

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