Reduced divisors and gonality in finite graphs [Bachelor thesis]

31 Pages·2012·0.507 MB·English

by  Josse van Dobben de Bruyn

Josse van Dobben de Bruyn
Reduced divisors and gonality in finite graphs
Bachelor’s thesis, 8 August 2012
Supervisor: Dr D.C. Gijswijt
Mathematisch Instituut, Universiteit Leiden
Contents
1 Introduction 2
2 Basic definitions 3
2.1 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 The Laplacian matrix and principal divisors . . . . . . . . . . . . . . . . . . . . . 4
2.4 Linear equivalence and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.5 Gonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Further techniques 7
3.1 Chip-firing games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Firing from subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Reduction of a divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 A lower bound on gon(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5 An upper bound on gon(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Examples 19
4.1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 Complete graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4 Complete bipartite graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.5 Two-dimensional grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5 Algorithmic approach 22
5.1 Divisor reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2 Rank of a divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.3 Gonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 Further reading 29
6.1 A conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
References 30
1
1 Introduction
In recent research, several results from the theory of Riemann surfaces have been related to
similar results on graphs. For instance, the Riemann–Roch theorem for Riemann surfaces has
an analogue for graphs, and the theory of (principal) divisors can also be applied to graphs. In
that fashion, gonality has been defined for graphs as well. It turns out that the gonality of a
graph is also related to winning strategies in certain chip-firing games on graphs, which have
been studied since the 1980s.
In this thesis, the most important results and techniques are constructed in Section 3, most
notably lower and upper bounds on the gonality. Some examples of graphs and their gonalities
can be found in Section 4.
Finally we will conduct an in-depth analysis of an algorithm for computing the reduced divisor
in Section 5. We also briefly discuss algorithms for other, related tasks, such as computing the
rank of a divisor and computing the gonality of a graph.
This thesis gives an overview of some of the recent developments in the area of graph gonality,
aimed at undergraduates in their final year as well as graduate students in mathematics. Some
prior knowledge of graph theory and group theory might be useful, but in general the text is
accessible for anyone with a general mathematical background (naive set theory, linear algebra).
2
2 Basic definitions
The theory of graph gonality is based upon various notions, which will briefly be discussed here.
2.1 Graph theory
Within this thesis, only finite, undirected, loopless multigraphs will be considered, which we
will assume to be non-empty and connected. That is, a graph is a triple G = (V,E,φ) of finite
disjoint sets V,E and a map φ : E → [V]2, where [V]2 denotes the set of 2-element subsets of
V. The elements of the set V are called vertices and the elements of E are called edges. By
definition, every edge e∈E maps to a set {a,b}⊂V consisting of exactly two distinct vertices,
called the ends of e. For every v ∈V, we let E(v) denote the set of edges incident with v, that
is: E(v)={e∈E :v ∈φ(e)}. Thedegreed(v)ofavertexv ∈V isthenumberofedgesincident
with v. In other words, we have d(v)=|E(v)|.
Throughout the majority of this text, we will only consider one graph at a time, which we will
simply denote by G. Moreover we will call the vertex set V, the edges E and the edge map φ.
2.2 Divisors
A key concept in the theory of gonality is the concept of divisors.
Definition 2.1. The set Div(G) of divisors on G is the set of all functions f : V → Z. This
becomes an abelian group by (f +g)(v):=f(v)+g(v).
Definition 2.2. The degree deg(D) of a divisor D ∈Div(G) is the sum of its entries:
(cid:88)
deg(D)= D(v).
v∈V
For convenience, we make three more definitions.
Definition 2.3. For any k ∈Z, the set Divk(G) is given by
Divk(G)={D ∈Div(G):deg(D)=k}.
NotethatDiv0(G)isasubgroupofDiv(G). Foranyotherk ∈Z, Divk(G)isacosetofDiv0(G).
Definition 2.4. We define a partial order ≤ on Div(G) by writing D ≤ D(cid:48) if and only if
D(v)≤D(cid:48)(v)holdsforallv ∈V. Thisuniquelydeterminesthepartialorders<,≥and>. Note
that D <D(cid:48) ⇐⇒ D ≤D(cid:48)∧D (cid:54)=D(cid:48), hence D <D(cid:48) doesn’t imply that D(v)<D(cid:48)(v) holds for
all v ∈V; strict inequality only has to hold for some v ∈V.
Definition 2.5. A divisor D ∈ Div(G) is said to be effective if D ≥ 0. The set containing all
effective divisors is denoted by Div (G). For any effective divisor D ∈Div (G), we let supp(D)
+ +
denote the support of D, that is:
supp(D)={v ∈V :D(v)>0}.
Definition 2.6. For all k ∈Z we define
Divk(G)=Divk(G)∩Div (G).
+ +
That is, Divk(G) is the set of all effective divisors of degree k on G.
+
3
2.3 The Laplacian matrix and principal divisors
One of the main elements of gonality theory is the Laplacian matrix of a graph, often simply
referred to as the Laplacian. The easiest way to define the Laplacian of G is in terms of two
other matrices A and D associated with G: D is the diagonal matrix given by D = d(v) for
v,v
all v ∈V, and A is the adjacency matrix of G, which is given by
 (cid:12) (cid:12)
 (cid:12)(cid:8)e∈E :φ(e)={v,w}(cid:9)(cid:12) if v (cid:54)=w,
(cid:12) (cid:12)
A =
v,w
 0 otherwise.
It is obvious from the definition that A is symmetric.
Definition 2.7. The Laplacian matrix Q of G is given by Q = D−A. The homomorphism
∆:Div(G)→Div(G) associated with the Laplacian matrix is called the Laplace operator.
Example 2.8. The complete bipartite graph K has the following Laplacian matrix:
2,3
v3  3 0 −1 −1 −1
v
1  0 3 −1 −1 −1
 
v4 −1 −1 2 0 0 .
 
v −1 −1 0 2 0 
2
−1 −1 0 0 2
v
5
NotethatthedrawingofQmaydependontheorderwechoosetorepresentthevertices,however
the abstract matrix Q is uniquely defined by the graph G. (cid:2)
Lemma2.9. Let1denotetheall-onesvector. TheLaplacianmatrixQ,whenviewedasamatrix
over Q, satisfies ker(Q)=span(1) and rank(Q)=|V|−1
Proof. As Q has zero row sums, we have 1∈ker(Q). For |V|=1 the result is obvious, for now
we have QV = span(1) ⊂ ker(Q). Assume |V| > 1 and let f ∈ QV be given with f ∈/ span(1).
We set m = max{f(v) : v ∈ V} and define U ⊂ V by setting U = {v ∈ V : f(v) = m}. Note
that U (cid:54)= V holds because f ∈/ span(1). Because G is connected, we can choose some u ∈ U
which has a neighbour w ∈V \U. Now we have A >0, so A ·f(w)<A ·m, hence
u,w u,w u,w
(cid:88)
(Qf)(u)=d(u)·f(u)− (A ·f(v))
u,v
v∈V
(cid:88)
>d(u)·f(u)−m· A
u,v
v∈V
=d(u)·m−m·d(u)=0.
Hence Qf (cid:54)= 0. This holds for all f ∈ QV \span(1). Therefore we have ker(Q) = span(1) and
rank(Q)=|V|−1, which concludes the proof.
Remark 2.10. Henceforth, we let Q denote the Laplacian matrix viewed as a matrix over any
fieldofcharacteristic0(inparticularQ). Ontheotherhand,∆willexclusivelybeusedtodenote
the homomorphism Div(G)→Div(G).
Definition2.11. Thedivisorsintheimageofthehomomorphism∆arecalledprincipaldivisors.
The subgroup of principal divisors is denoted by Prin(G). In other words, we have
Prin(G)=∆[Div(G)].
4
Since every column of Q has total sum 0, every principal divisor has degree 0, so we have
Prin(G)⊂Div0(G).
Since both Prin(G) and Div0(G) are subgroups of Div(G), the above inclusion tells us that
Prin(G) is in fact a subgroup of Div0(G). This leads us to the next definition.
Definition2.12. TheJacobianJac(G)ofGisdefinedtobethequotientgroupDiv0(G)/Prin(G).
Theorem 2.13. Let κ(G) denote the number of spanning trees in G. The group Jac(G) is finite
of order κ(G).
Proof. Choose some v0 ∈V. Now we define a map ϕ:ZV\{v0} →Div0(G) by

D(v) if v ∈V \{v },
 0
ϕ(D)(v)= (cid:88)
− D(w) if v =v .
 0
w(cid:54)=v0
That is, we set φ(D)(v ) such that deg(φ(D))=0 holds. This is obviously an isomorphism. Let
0
Q bethematrixobtainedfromQbydeletingthev -throwandcolumn. Nowthe|V|−1columns
v0 0
of Qv0 span a lattice Λ ⊂ ZV\{v0} of maximal rank. It follows that det(Qv0) = [ZV\{v0} : Λ].
On the other hand, it follows from the matrix tree theorem that det(Q ) = κ(G). Moreover,
v0
the deleted column is an integer multiple of the other columns (since G is connected), hence we
have ϕ[Λ]=Prin(G). Therefore we have [Div0(G):Prin(G)]=[ZV\{v0} :Λ]=κ(G).
2.4 Linear equivalence and dimension
We define a relation ∼ on Div(G) by having D ∼D(cid:48) if and only if D−D(cid:48) ∈Prin(G). Note that
this is an equivalence relation. This is often referred to as linear equivalence (e.g. in [2]), hence
we will use that terminology as well.
Definition 2.14. The linear system associated to a divisor D ∈Div(G) is defined to be the set
|D| containing all effective divisors that are equivalent to D, that is:
|D|={D(cid:48) ∈Div (G):D ∼D(cid:48)}.
+
TheequivalenceclassofadivisorD ∈Div(G)(withrespecttolinearequivalence)willbedenoted
by [D]. Note that |D|=[D]∩Div (G).
+
Definition 2.15. The dimension r(D) of the linear system |D| is defined to be
r(D)=max(cid:8)s∈Z:|D−F|=(cid:54) ∅ for all F ∈Divs(G)(cid:9).
+
Remark 2.16. Note that the maximum is always attained, since r(D) ≤ deg(D) holds. That
is, for every F ∈Div (G) with deg(F)>deg(D) we have deg(D−F)<0, hence |D−F|=∅.
+
Moreover, observe that r(D)=−1 if and only if |D|=∅.
Remark 2.17. Neither the linear system |D| nor its dimension r(D) depend on the choice of
its representative D. That is, if D ∼ D(cid:48), then obviously we have |D| = |D(cid:48)|. Moreover, for all
F ∈Div(G) we have D−F ∼D(cid:48)−F, hence r(D)=r(D(cid:48)).
5
2.5 Gonality
As a special case of Remark 2.16, we have deg(D) ≥ 1 for all D ∈ Div(G) with r(D) ≥ 1.
Moreover, note that in every graph there is a divisor D ∈ Div(G) such that r(D) ≥ 1; consider
for instance the all-ones divisor D =1. Now we can make the following definition.
Definition 2.18. The gonality gon(G) of a graph G is defined by
gon(G)=min{deg(D):D ∈Div(G), r(D)≥1}.
Thisminimumisguaranteedtoexist, astheset{deg(D):D ∈Div(G), r(D)≥1}isnon-empty
and bounded from below by 1.
Corollary 2.19. For any graph G we have gon(G)≥1. (cid:3)
Example 2.20. Let G be a tree. By Theorem 2.13 the Jacobian Jac(G) is trivial, so we have
Prin(G) = Div0(G). Thus for any D,D(cid:48) ∈ Div(G) with deg(D) = deg(D(cid:48)) we have D ∼ D(cid:48).
In particular, for any D,D(cid:48) ∈ Div1(G) we have 0 ∈ |D − D(cid:48)|, so r(D) ≥ 1. Now observe
+
that Div1(G) is not empty as for every v ∈ V the unit divisor e , which is one in v and zero
+ v
elsewhere, is an element of Div1(G). Therefore we have gon(G)≤1. By Corollary 2.19 we have
+
gon(G)≥1, hence gon(G)=1 must hold. (cid:2)
More examples will follow in Section 4. We will close this section with the following lemma,
which is actually quite surprising. As it turns out, we can limit ourselves to effective divisors D
in Definition 2.18. This is not immediately obvious from the definition, but it severely reduces
the number of divisors we have to consider if we attempt to determine gon(G).
Lemma 2.21. There exists some D ∈Div (G) such that deg(D)=gon(G) and r(D)≥1.
+
Proof. LetD(cid:48) ∈Div(G)beadivisorsuchthatdeg(D(cid:48))=gon(G)andr(D(cid:48))≥1. Nowfixv ∈V.
Since r(D(cid:48))≥1, there is some effective divisor F ∈Div(G) such that D(cid:48)−e ∼F. But now we
v
haveD(cid:48)−e −F ∈Prin(G), soitfollowsthatD(cid:48) ∼F+e . SincebothF ande areeffective, so
v v v
isF+e . ThuswesetD =F+e . Obviously,wehaveD(cid:48) ∼Danddeg(D)=deg(D(cid:48))=gon(G).
v v
Moreover, we can conclude from D ∼D(cid:48) that r(D)=r(D(cid:48))≥1.
Corollary 2.22. We have gon(G)=min{deg(D):D ∈Div (G), r(D)≥1}. (cid:3)
+
6
3 Further techniques
3.1 Chip-firing games
The abstract concepts of the previous chapter can also be viewed in the context of chip-firing
games. Inachip-firinggame,weconsideragraphGwithanyintegernumberofchips(commonly
dollars, although other currencies are not at all uncommon) on its vertices, which can be moved
aboutfollowingcertainrules. IntheNorine–Bakerchip-firinggame([2]),therulesareasfollows:
1. The game is played by two people, you and an evil adversary. A graph G is given.
2. First, you have to divide a number of chips among the vertices of G. Vertices are allowed
to have a negative number of chips; such vertices are said to be in debt.
3. The opponent then chooses any one vertex and subtracts one chip from that vertex. This
vertex may already be in debt.
4. Subsequently, you are allowed to move the chips about. However, there is only one type of
move you can do: you can move deg(v) chips from node v to all its neighbours by giving
each of them 1 chip. You are not allowed to leave certain neighbours out or give some
of them multiple chips! Using a sequence of moves of this type (called firing moves or
borrowing moves), you have to make sure that all vertices are out of debt.
5. If you succeed, you win. Otherwise, the adversary wins.
We assume that the opponent plays his optimal strategy: he will win whenever he can. Thus,
whether or not there will be a winning strategy for the remainder of the game depends on the
choice of initial conguration in step 2.
Note that we can denote the state of G (that is, the number of chips on the vertices of G) at
any moment during the game as a divisor on G. Now we have the following lemma, leading to
an important theorem.
Lemma3.1. Thefiringmovescorrespondtosubtractingprincipaldivisors,andtheinitialdivisor
isequivalenttotheresultingdivisor. Moreover,everyequivalentdivisorcanbereachedviaafinite
sequence of firing moves.
Proof. Let D be the initial divisor. Let e be the unit divisor corresponding to some vertex
v
v ∈ V, then −Qe is the vector containing −deg(v) on the v-th coordinate, +1 on each of the
v
neighbours of v and 0 on all other vertices. Thus D−Qe is the divisor resulting from firing
v
from vertex v to all of its neighbours.
Inordertoseewhyeveryequivalentdivisorcanbereachedusingafinitesequenceoffiringmoves,
let D be the initial divisor and D an equivalent divisor. There exists some f ∈ Div(G) such
1 2
that D = D −∆f, however the entries of f do not have to be positive. Now if we recall that
2 1
1∈ker(∆),weseethatwecanaddanyconstantdivisork·1tof inordertoobtainf(cid:48) ∈Div (G)
+
which also satisfies D =D −∆f(cid:48). Now f(cid:48) corresponds to a finite sequence of firing moves, so
2 1
every equivalent divisor can in fact be reached using a finite sequence of firing moves.1
1The point is that the game only allows a positive number of firing moves on each vertex, whereas principal
divisors may arise from a negative number of firing moves. The above argument assures us that even those
principaldivisorscanbeachievedwithintherulesofthechip-firinggame.
7
Theorem 3.2. The gonality of G equals the minimum number of chips in a winning initial
configuarion in step 2 of the Norine–Baker chip-firing game.
Proof. Assume that the game is winning for us if we start out with some divisor D ∈ Div(G).
Subsequently, the opponent subtracts one chip from a vertex he chooses. Regardless of the chip
he subtracts, there is a winning strategy for us (for the remainder of the game). That is: for
every F ∈Div1(G), there is some effective divisor F(cid:48) ∈Div(G) such that D−F ∼F(cid:48). In other
+
words: |D−F|=(cid:54) ∅. Hence we have r(D)≥1.
On the other hand, if the game is not winning for us after choosing some initial configuration
D ∈ Div(G), then there must be some F ∈ Div1(G) such that |D−F| = ∅, so now we have
+
r(D) < 1. All in all, the game is winning after choosing some initial divisor D ∈ Div(G) if and
only if r(D)≥1.
As a result, we can think of gonality in terms of this chip-firing game. This helps our under-
standing of gonality and provides us with a useful toolkit for proving lemmata and theorems.
3.2 Firing from subsets
Apart from firing from a single vertex, one might also consider firing once from every vertex in
some subset A⊂V. We will denote the indicator function of A as 1 , that is:
A
(cid:40)
1 if v ∈A,
1 (v)=
A
0 if v ∈/ A.
Thus, henceforth the process of starting with some initial configuration D ∈ Div(G) and firing
from a subset A ⊂ V will be denoted by D−∆1 . When firing from a subset A, the edges
A
between two vertices v,w ∈ A will have one chip going from v to w and one chip going from w
to v, so they cancel out. Hence we only have to worry about edges leaving A.
Definition 3.3. LetA⊂V beasubsetofvertices. Theoutdegreeoutdeg (v)ofavertexv ∈A
A
with respect to A is the number of edges leaving A through v, that is:
(cid:12) (cid:12)
outdeg (v)=(cid:12)(cid:8)e∈E :φ(e)∩A={v}(cid:9)(cid:12).
A (cid:12) (cid:12)
Moreover, we let outdeg (A) denote the total outdegree of A. Similarly, the indegree indeg (v)
A A
of a vertex v ∈V \A with respect to A is the number of edges entering V \A through v:
(cid:12) (cid:12)
indeg (v)=(cid:12)(cid:8)e∈E :φ(e)∩(V \A)={v}(cid:9)(cid:12).
A (cid:12) (cid:12)
Remark 3.4. Note that the only difference is that outdeg only applies to vertices in A whereas
indeg only applies to vertices in V \A. That is, for all v ∈A we have outdeg (v)=indeg (v)
A V\A
and for all v ∈V \A we have indeg (v)=outdeg (v).
A V\A
Example3.5. AssumethatD =D −∆1 . Forallv ∈AwehaveD (v)=D (v)−outdeg (v)
2 1 A 2 1 A
and for all v ∈V \A we have D (v)=D (v)+indeg (v). It is now apparent from ∆1=0 that
2 1 A
firingfromasubsetAcanbeundonebysubsequentlyfiringfromV \A. Inparticular, onecould
reverse fire into a vertex v ∈ V by firing from every vertex in V \{v}. Starting with an initial
divisor D ∈Div(G), we then end up with the divisor D(cid:48) =D+∆e . This is also the main idea
v
of the proof of Lemma 3.1. (cid:2)
8
Example 3.6. Let G be a tree and let D ∈ Div1(G) be the effective divisor of degree 1 given
+
by D = e for some v ∈ V. For any neighbour w of v, the edge e with φ(e) = {v,w} induces a
v
cut(U,V \U)separatingv andw. ByfiringfromU,wesendonechipfromv tow. Allthefiring
moves on interior edges of U get leveled out by another firing move in the opposite direction, so
the only thing that changes is the chip traveling from v to w. This corresponds with the finding
that Jac(G) is trivial for any tree. (cid:2)
A major advantage of considering firing from subsets rather than firing from single vertices
becomes apparent when we look at level sets.
Definition 3.7. Let D,D(cid:48) ∈ Div(G) be two equivalent divisors with D(cid:48) = D−∆f for some
f ∈ Div(G). Define m = max{f(v) : v ∈ V} and k = m−min{f(v) : v ∈ V}. The level set
decomposition of f is the sequence of sets A ⊂A ⊂···⊂A =V given by
0 1 k
A ={v ∈V :f(v)≥m−i}.
i
The sequence of divisors D ,D ,...,D ∈[D] given by D =D and D =D −∆1 for all
0 1 k 0 i+1 i Ai
i∈{0,1,...,k−1} is called the divisor sequence associated with the level set decomposition.
Remark 3.8. Note that f is not uniquely determinded as we are free to add or subtract any
integer multiple of 1, but the choice of f does not affect the level sets. Therefore we will usually
assumethatf(v)≥0holdsforallv ∈V insuchawaythatf(v )=0holdsforatleastonev ∈V.
0 0
In that case we have k =m, so for all i∈{0,1,...,k} we have A ={v ∈V :f(v)≥k−i}.
i
Lemma 3.9. For all i∈{0,1,...,k−1} we have A (cid:54)=V.
i
Proof. By the definition of k, there is some v ∈ V such that f(v) = m−k. Moreover, for all
i∈{0,1,...,k−1} and all w ∈A we must have f(v)≥m−i>m−k, hence v ∈/ A .
i i
Theorem 3.10. Let D,D(cid:48) ∈ Div(G) be two equivalent divisors with D(cid:48) = D −∆f for some
f ∈Div(G). Let A ⊂A ⊂···⊂A be the level set decomposition of f and D ,D ,...,D the
0 1 k 0 1 k
associated divisor sequence. Then for all i ∈ {0,1,...,k} we have D (v) ≥ min(cid:0)D(v),D(cid:48)(v)(cid:1)
i
(for all v ∈V). Moreover, we have D =D(cid:48).
k
Proof. Define f ∈Div(G) recursively by f =f and f =f −1 . Inductively, it is not hard
i 0 i+1 i Ai
to see that D(cid:48) =D −∆f holds for all i∈{0,1,...,k}.
i i
• For i=0, we have D =D and f =f, so we have D(cid:48) =D−∆f =D −∆f .
0 0 0 0
• Leti∈{0,1,...,k−1}begivensuchthatD(cid:48) =D −∆f . NownotethatD =D −∆1
i i i+1 i Ai
and f =f −1 , so we find
i+1 i Ai
D(cid:48) =D −∆f =D −∆(f +1 )=D −∆1 −∆f =D −∆f .
i i i i+1 Ai i Ai i+1 i+1 i+1
Hence the property holds for j =i+1 as well.
Moreover, we will prove (once again by induction) that for all i ∈ {0,1,...,k}, any v ∈ A
i
satisfies f (v)=m−i, whereas any v ∈V \A satisfies f (v)<m−i.
i i i
• For i=0, this follows from the definition of A .
i
• Leti∈{0,1,...,k−1}begivensuchthatf (v)=m−iholdsforallv ∈A andf (v)<m−i
i i i
holds for all v ∈V \A . Let v ∈V be given. We distinguish three cases.
i
9