Frobenius number, Riemann-Roch structure, and zeta functions of graphs

18 Pages·2008·0.255 MB·English

by  Dino Lorenzini

FROBENIUS NUMBER, RIEMANN-ROCH STRUCTURE, AND ZETA
FUNCTIONS OF GRAPHS
DINO LORENZINI
We investigate in this article possible generalizations of the Riemann-Roch theorem
for graphs of Baker and Norine [2]. We propose such a generalization for certain integer
lattices of rank n − 1 in Zn. To a lattice endowed with a Riemann-Roch structure we
associate a two-variable Riemann-Roch zeta-function with a functional equation, and we
investigate the properties of this function.
1. Generalization of the Frobenius number
Let R ∈ Zn be a vector with strictly positive integers entries. We denote its transpose
by tR = (r ,...,r ). In this article, unless specified otherwise, any integer vector denoted
1 n
R will be assumed to have gcd(r ,...,r ) = 1.
1 n
Let D ∈ Zn. We define the degree of D as deg (D) := D·R. When the context makes
R
the reference to R unnecessary, we will denote deg simply by deg. The kernel of the
R
degree homomorphism Zn → Z is the lattice in Zn perpendicular to R:
Λ := {D ∈ Zn,D·R = 0}.
R
For any sublattice Λ ⊆ Λ of rank n − 1, we define Pic(Λ) := Zn/Λ. If D ∈ Zn, we
R
denote by [D] the class of D in Pic(Λ). By construction, deg(Λ) = {0}, so that we have
a group homomorphism
deg : Pic(Λ) → Z, deg([D]) := D·R.
The kernel of this homomorphism is the finite abelian group
Pic0(Λ) := Λ /Λ.
R
If D ∈ Zn, we will write D ≥ 0 if all coefficients of D are non-negative, and we write
D > 0 if all coefficients of D are strictly positive. Note that if D ≥ 0, then deg(D) ≥ 0.
We will say that D ≥ 0 is effective. An element D ∈ Zn may be called a divisor, and
[D] ∈ Pic(Λ) a divisor class, in keeping with the notation used in the Riemann-Roch
theorem for curves.
1.1 Let us define a new invariant of Λ, the g-integer of Λ, as follows. Consider a non-
negative integer γ such that, for any vector D ∈ Zn such that deg(D) ≥ γ, then there
exists E ≥ 0 with D−E ∈ Λ (or, in other words, such that [D] = [E]). We let g = g(Λ)
be the smallest such integer γ. We show in 1.4 that the integer g(Λ) exists.
1.2 Fixpositiveintegersr ,...,r . Wedefineg(r ,...,r )tobeonemorethanthelargest
1 n 1 n
integer that does not belong to the additive semigroup of Z generated by r ,...,r . In
1 n
other words, every integer N ≥ g(r ,...,r ) can be written in the form Pn x r with
1 n i=1 i i
x ≥ 0 for all i = 1,...,n, and g(r ,...,r )−1 cannot be written in this form. The integer
i 1 n
g(r ,...,r )−1 is called the Frobenius number of r ,...,r in the literature. We will call
1 n 1 n
g(r ,...,r ) the Frobenius number in this article; we chose this rescaling so that we have
1 n
the property g(r ,...,r ) ≥ 0. In particular, if r = 1 for some i, g(r ,...,r ) = 0.
1 n i 1 n
Date: February 28, 2008.
1
FROBENIUS NUMBER, RIEMANN-ROCH STRUCTURE, AND ZETA FUNCTIONS OF GRAPHS 2
Lemma 1.3. Let R > 0 be as above, and Λ0 ⊆ Λ ⊆ Λ be lattices of rank n−1.
R
a) g(Λ ) = g(r ,...,r ) ≥ 0, and g(Λ ) 6= 1. In particular, if |Pic0(Λ)| = 1, then
R 1 n R
Λ = Λ , and g(Λ) = g(r ,...,r ).
R 1 n
b) If Λ0 ⊆ Λ ⊆ Λ , then we have a natural surjective homomorphism Pic(Λ0) → Pic(Λ),
R
and g(Λ0) ≥ g(Λ) ≥ g(Λ ).
R
c) If g(Λ) = 0, then Λ = Λ .
R
Proof. (a) By construction, Pic0(Λ ) = (1). Thus, if there exists E ≥ 0 of degree d, every
R
D ∈ Zn of degree d is such that [D] = [E] in Pic(Λ ). And if there is no E ≥ 0 of degree
R
d, then no element D ∈ Zn of degree d is equivalent to an effective. This shows that
g(Λ ) = g(r ,...,r ). By definition, g(r ,...,r ) 6= 1.
R 1 n 1 n
(b) Follows from the definitions.
(c) If g(Λ) = 0, we find that every D of degree 0 is equivalent to an effective. But there
is only one effective E ≥ 0 with deg(E) = 0, the zero vector. Hence Pic0(Λ) = (1), and
Λ = Λ . (cid:3)
R
Our next proposition implies that the integer g(Λ) exists. Given any positive integer
x, we let xΛ := {xD,D ∈ Λ}. Denote by e = e(Λ) the exponent of the group Pic(Λ). In
particular, when Λ ⊆ Λ , eΛ ⊆ Λ.
R R
Proposition 1.4. Let R > 0 be as above, and Λ ⊆ Λ be a lattice of rank n−1. Then
R
g(Λ) ≤ eg(Λ )+(e−1)(−1+Pn r ), and e(Λ) ≥ g(Λ)−1+Pni=1ri .
R i=1 i g(ΛR)−1+Pni=1ri
Proof. The second inequality is immediate from the first. To prove the first, we note that
since eΛ ⊆ Λ, g(Λ) ≤ g(eΛ ). We claim that g(eΛ ) ≤ eg(Λ )+(e−1)(−1+Pn r ).
R R R R i=1 i
Indeed, let D ∈ Zn. Write D = eD0 + t(y ,...,y ) with 0 ≤ y ≤ e − 1. Suppose
1 n i
that deg(D) > e(g(Λ)−1)+(e−1)(Pn r ). Then deg(eD0) > e(g(Λ)−1). It follows
i=1 i
that deg(D0) ≥ g(Λ). Hence, there exists E0 ≥ 0 and V0 ∈ Λ such that D0 = E0 + V0.
Thus, D = eE0 + t(y ,...,y ) + eV0, and D is eΛ-equivalent to an effective. Hence,
1 n
g(eΛ) ≤ 1+e(g(Λ)−1)+(e−1)(Pn r ). (cid:3)
i=1 i
In view of the fact that g(Λ ) = g(r ,...,r ), we can interpret the integer g(Λ) as a
R 1 n
generalization of the Frobenius number to lattices.
1.5 LetGbeafiniteunweightedconnectedmultigraphonnverticesandmedges, without
loop edges. Choose an ordering v ,...,v for the vertices of G. Let d denote the valency
1 n i
of v . Let A denote the associated adjacency matrix. Set D := diag(d ,...,d ), the
i 1 n
diagonal matrix of the valencies. Let M := D − A, the Laplacian matrix of G. Set
Λ = Im(M).
G
The order of the group Pic0(Λ ) is well-known to be the number κ(G) of spanning
G
trees of G ([5], 6.3). The group Pic0(Λ ) occurs in the literature under different names,
G
depending on the context in which it is used: group of components [16] (1989), sandpile
group [9] (1990), jacobian group [1] (1997), or critical group [6] (1999). See [19] for the
relationships between this group and the eigenvalues of the Laplacian. Recall that the
integer β(G) := m−n+1 is the first Betti number of the graph. The work of Baker and
Norine [2] completely determines the integer g(Λ ).
G
Proposition 1.6. Let G be a graph as above. Then g(Λ ) = m−n+1.
G
Proof. For any D ∈ Zn, Baker and Norine introduce an integer r(D) with the following
property([2],2.1): r(D) ≥ −1,andr(D) > −1ifandonlyifDisequivalenttoaneffective.
Theorem 1.12 in [2] states that for all D ∈ Zn, r(D)−r(K −D) = deg(D)+1−β(G).
Then deg(D)+1−β(G) = r(D)−r(K −D) ≤ r(D)+1. Assume that deg(D) ≥ β(G).
FROBENIUS NUMBER, RIEMANN-ROCH STRUCTURE, AND ZETA FUNCTIONS OF GRAPHS 3
Then deg(D)−β(G) ≥ 0, so that r(D) ≥ 0. It follows that g(Λ ) ≤ β(G). A complete
G
description of the set of D ∈ Zn of degree β(G)−1 which are not equivalent to an effective
is given in [2], 3.4, showing in particular that this set is not empty. Hence, g(Λ ) = β(G).
G
(cid:3)
Let e denote the exponent of the group Pic0(Λ ). Then 1.4 shows that e ≥ m . This
G n−1
bound is achieved when G is a graph on two vertices linked by m edges. Note that when
G has vertex connectivity at least 2, κ(G) ≥ m ([19], 4.3).
1.7 Let Λ ⊆ Λ be a lattice of rank n−1. Let D ∈ Zn. Set (cid:15) (D) to be 1 if there exists
R Λ
E ≥ 0 such that [D] = [E] in Pic(Λ), and set (cid:15) (D) to be 0 if there does not exist any
Λ
E ≥ 0 such that [D] = [E] in Pic(Λ).
A canonical vector K ∈ Zn for Λ is a vector of degree deg(K) = 2g(Λ)−2 such that,
for all D ∈ Zn of degree g(Λ)−1,
(cid:15) (D) = (cid:15) (K −D).
Λ Λ
The Riemann-Roch theorem of Baker and Norine [2] shows that a canonical vector exists
for the lattice Λ associated to any graph G.
G
Proposition 1.8. Let G be a graph as above, and let Λ ⊆ Zn be its associated Laplacian
G
lattice. Let K := (d −2,...,d −2). Then deg(K) = 2g(Λ )−2 and K is a canonical
1 n G
vector for Λ .
G
Proof. The condition (cid:15) (D) = (cid:15) (K − D) for all D of degree g(Λ) − 1 is equivalent to
Λ Λ
condition RR2 in [2], 2.2, and is proven to hold in the proof of 1.12 of [2]. (cid:3)
The motivation for introducing the notions of g-number and of canonical vector is
found in 3.1 and 3.4. The existence of a canonical vector for Λ allows for the existence
of a Riemann-Roch structure on Λ, to which one associates a two-variable zeta-function.
These topics are treated in the third section of this article. In the next section, we
investigate the existence of a canonical vector for lattices Λ that are not of the form Λ .
G
Let us mention here two other questions that are not yet settled.
1) Is it possible to find Λ0 ( Λ with g(Λ0) = g(Λ)?
2) Is it possible to find Λ with two canonical vectors K and K0 such that [K] 6= [K0] in
Pic(Λ)?
1.9 Let d(m,r ,...,r ) denote the number of non-negative integer solutions of the equa-
1 n
P
tion m = x r , with x ≥ 0 for all i = 1,...,n. It is well-known that
i i i
∞
1 X
= d(m,r ,...,r )tm.
(1−tr1)·...·(1−trn) 1 n
m=0
Let M (x) denote the largest integer m such that d(m,r ,...,r ) = x, when it exists.
R 1 n
When d(m,r ,...,r ) 6= x for all m ∈ Z, we set M (x) = 0. By definition, M (0) =
1 n R R
g(r ,...,r )−1.
1 n
Lemma 1.10. Fix n and R, and consider M (x).
R
a) Let Λ ⊆ Λ be a lattice of rank n−1. Then
R
Max{M (x),x ∈ [0,|Pic0(Λ)|−1]} < g(Λ).
R
b) Given e ∈ N, Max{M (x),x ∈ [0,en−1 −1]} ≤ e(g(R)−1)+(e−1)(Pn r ).
R i=1 i
FROBENIUS NUMBER, RIEMANN-ROCH STRUCTURE, AND ZETA FUNCTIONS OF GRAPHS 4
Proof. (a) If m ≥ g(eΛ), then d(m,r ,...,r ) ≥ |Pic0(Λ)|, since there must exist at least
1 n
|Pic0(Λ)| distinct effective E ≥ 0 of degree m.
(b) Consider the lattice eΛ , with Pic0(eΛ ) = Λ /eΛ of order en−1. Apply (a), with
R R R R
the formula for g(Λ ) from 1.4. (cid:3)
R
It is well-known that when n = 2, M (x) = (x+1)r r −r −r . This is attributed in
R 1 2 1 2
[10], page 65, to A. D. Wheeler in 1860. We then have an equality
Max{M (x),x ∈ [0,e−1]} = M (e−1) = e(g(r ,r )−1)+(e−1)(r +r ).
R R 1 2 1 2
When n = 3, Max{M (0),M (1),M (2),M (3)} ≤ 2M (0)+r +r +r , with equality
R R R R R 1 2 3
achieved, for instance, when (r ,r ,r ) = (3,4,12) or (5,6,30). Thanks to Brian Cook
1 2 3
for pointing out to us that M (4) = 2M (0)+r +r +r +1 when (r ,r ,r ) = (3,4,5)
R R 1 2 3 1 2 3
or (3,7,8). We do not know whether the inequality M (4) ≤ 2M (0)+r +r +r +1
R R 1 2 3
holds in general.
2. Existence of a canonical vector
In this section, we introduce several classes of lattices Λ of rank n−1 for which we can
compute g(Λ) and show the existence of a canonical vector. Let us note first the following
easy facts.
Lemma 2.1. Let Λ ⊆ Λ be a lattice of rank n−1.
R
a) Assume that Pic0(Λ) has at most two distinct classes that contain an effective divisor,
or at most two classes that do not contain any effective divisor. Then Λ has a canonical
vector K. In particular, if |Pic0(Λ)| ≤ 5, then Λ has a canonical vector K.
b) If g(Λ) ≤ 1, then Λ has a canonical vector K, and the class of K in Pic0(Λ) is uniquely
determined.
Proof. (a) When |Pic0(Λ)| = 1, any vector K of degree 2g(R)−2 is a canonical vector.
If all divisors D of degree g(Λ)−1 are not equivalent to an effective, then any divisor of
degree 2g(Λ)−2 is a canonical vector. If [D] is the only class that is either equivalent to
an effective, or not equivalent to an effective, then 2D is a canonical vector. If [D] and
[D0] are the only classes that are equivalent to an effective, or if they are the only classes
that are not equivalent to an effective, then we can take D+D0 as a canonical vector.
(b) When g(Λ) = 0, |Pic0(Λ)| = 1, and the result follows from (a). When g(Λ) = 1,
the 0-vector (0,...,0) is the unique vector of degree g(Λ) − 1 that is effective. Then
K = (0,...,0) is a canonical vector. If K0 is another canonical vector, 0 and K0−0 both
need to be effective, so that K0 −0 is equivalent to 0, and [K0] = [0]. (cid:3)
Remark 2.2 Let us note that if Pic0(Λ) is killed by 2 and there are exactly two distinct
classes [D] and [D0] of degree g(Λ)−1 which are not equivalent to an effective, then Λ has
two canonical vectors that are not equivalent. Indeed, 2[D−D0] = [0] in Pic0(Λ). Taking
K := 2D, we find that K − D = D and [K − D0] = [D0], so K is a canonical vector.
Taking now K0 := D +D0, we find first that K −D = D0 and K −D0 = D, so that K0
is a canonical vector. We have [K] 6= [K0], since otherwise, [2D] = [D +D0] implies that
[D] = [D0], a contradiction.
2.3 Let R > 0 and consider Λ ⊆ Λ , a lattice of rank n−1. Let x ,...,x be positive
R 1 n
integers, and let ‘ := lcm(x ,...,x ). Let X := diag(x ,...,x ), and consider the map
1 n 1 n
X : Zn → Zn. Let XΛ denote the image of Λ under the map X. Clearly, XΛ has
rank n − 1. Let d := gcd(‘r /x ,...,‘r /x ). Set R0 := t(‘r /dx ,...,‘r /dx ). Then
1 1 n n i 1 n n
XΛ ⊆ Λ .
R0
FROBENIUS NUMBER, RIEMANN-ROCH STRUCTURE, AND ZETA FUNCTIONS OF GRAPHS 5
We claim that d | ‘. Indeed, let p be prime, and assume that ps is the exact power of p
that divides d. Since gcd(r ,...,r ) = 1, there exists r with p - r . Hence, ps | ‘/x , and
1 n i i i
so ps | ‘.
Let J denote the transpose of the vector (1,...,1). As usual, I denotes the (n×n)-
n n
identity matrix. Given a lattice Λ ⊆ Λ , let
R
N(Λ) := {[D] ∈ Pic(Λ) | deg ([D]) = g(Λ)−1,(cid:15) (D) = 0}.
R Λ
Note that a vector K of degree 2g(Λ)−2 is a canonical vector for Λ if and only if, for all
[D] ∈ N(Λ), [K −D] ∈ N(Λ).
Proposition 2.4. Let Λ ⊆ Λ . Let X and R0 be as above, and consider XΛ ⊆ Λ .
R R0
a) g(XΛ)−1 = ‘(g(Λ)−1)+Pn (x −1)‘r /dx .
d i=1 i i i
b) The map N(Λ) → N(XΛ), [D] 7→ [XD+ t(x −1,...,x −1)], is a bijection.
1 n
c) If K is a canonical vector for Λ, then XK+2(X−I )J is a canonical vector for XΛ.
n n
d) If K(XΛ) is a canonical vector for XΛ, then K(XΛ) is XΛ-equivalent to XK+2(X−
I )J with K a canonical vector for Λ.
n n
Proof. (a) Write D0 ∈ Zn as D0 = XD +t(y ,...,y ) with 0 ≤ y ≤ x − 1 for all
1 n i i
i = 1,...,n. If deg (D0) > ‘(g(Λ) − 1) + Pn (x − 1)‘r /dx , then deg (XD0) >
R0 d i=1 i i i R0
‘(g(Λ) − 1) and, hence, deg (D) > g(Λ) − 1. It follows that D is Λ-equivalent to an
d R
effective, so that XD and D0 are XΛ-equivalent to an effective. Therefore, g(XΛ)−1 ≤
‘(g(Λ)−1)+Pn (x −1)‘r /dx .
d i=1 i i i
Assume now that deg (D0) = ‘(g(Λ) − 1) + Pn (x − 1)‘r /dx . If (y ,...,y ) 6=
R0 d i=1 i i i 1 n
(x −1,...,x −1), we find that deg (XD) > ‘(g(Λ)−1) and conclude as before that
1 n R0 d
D0 is XΛ-equivalent to an effective.
If (y ,...,y ) = (x −1,...,x −1), then deg (XD) = ‘(g(Λ)−1) and deg (D) =
1 n 1 n R0 d R
g(Λ) − 1. We claim that D0 is XΛ-equivalent to an effective if and only if D is Λ-
equivalent to an effective. It is clear that if D is Λ-equivalent to an effective, then D0
is XΛ-equivalent to an effective. Suppose now that D has degree g(Λ)−1, and is such
that there does not exist V ∈ Λ and E ≥ 0 such that D = V + E. Consider D0 :=
XD+ t(x −1,...,x −1). If there exist E0 ≥ 0 and XV0 ∈ XΛ such that D0+XV0 = E0,
1 n
we find that X(D+V0)+ t(x −1,...,x −1) ≥ 0. This can only occur if all coefficients
1 n
of D0 + V are non-negative, which contradicts our assumption on D. Since there exist
a D of degree deg (D) = g(Λ)−1 that is not Λ-equivalent to an effective, we find that
R
g(XΛ) = 1+ ‘(g(Λ)−1)+Pn (x −1)‘r /dx .
d i=1 i i i
(b) Note first that the map N(Λ) → N(XΛ), [D] 7→ [XD + t(x −1,...,x −1)], is
1 n
well-defined. We have shown in the proof of (a) that if [D] ∈ N(Λ), then [XD + t(x −
1
1,...,x −1)] ∈ N(XΛ). Moreover, if [D ] = [D], then [XD + t(x −1,...,x −1)] =
n 1 1 1 n
[XD+ t(x −1,...,x −1)], since D −D ∈ Λ implies that XD −XD ∈ XΛ. The map
1 n 1 1
is injective, since if [XD + t(x −1,...,x −1)] = [XD+ t(x −1,...,x −1)] for some
1 1 n 1 n
D and D, we have XD −XD ∈ XΛ, which implies that D −D ∈ Λ. We proved the
1 1 1
surjectivity of the map in (a), since we showed that every D0 that is not XΛ-equivalent
to an effective is of the form XD + t(x −1,...,x −1) for some D not Λ-equivalent to
1 n
an effective.
(c)Byhypothesis, themapN(Λ) → N(Λ), [D] 7→ [K−D], iswell-definedandbijective.
We leave it to the reader to check that this implies that the map N(XΛ) → N(XΛ),
[D0] 7→ [XK +2t(x −1,...,x −1)−D0], is well-defined and bijective.
1 n
(d) Let K0 be a canonical vector for K(XΛ). Then the map N(XΛ) → N(XΛ),
[D0] 7→ [K0−D0], is well-defined and bijective. Starting with any [D ] ∈ N(Λ), we obtain
0
that there exists [D] ∈ N(Λ) such that [K0−XD − t(x −1,...,x −1)] = [XD+ t(x −
0 1 n 1
FROBENIUS NUMBER, RIEMANN-ROCH STRUCTURE, AND ZETA FUNCTIONS OF GRAPHS 6
1,...,x −1)]. It follows that K0 is XΛ-equivalent to X(D +D)+2t(x −1,...,x −1).
n 0 1 n
We leave it to the reader to show that D +D is a canonical vector for Λ. (cid:3)
0
Corollary 2.5. Let G be a graph with n vertices, m edges, and adjacency matrix A.
Let D = diag(d ,...,d ) be the diagonal matrix of the valencies of the vertices, and let
1 n
M := D −A. Let Λ := Im(M) ⊆ Λ , with canonical vector K = t(d −2,...,d −2)
G Jn 1 n
and |Pic0(Λ )| = κ(G). Then
G
a) g(XΛ ) = 1+‘m−Pn ‘/x .
G i=1 i
b) The lattice XΛ has canonical vector t(x d −2,...,x d −2).
G 1 1 n n
c) The group Pic0(XΛ ) has order κ(G)(Qn x )/‘.
G i=1 i
Proof. Since gcd(‘/x ,...,‘/x ) = 1, we may apply the previous proposition to obtain
1 n
(a) and (b) with R0 = t(‘/x ,...,‘/x ) and g(Λ ) = m−n+1.
1 n G
The adjoint matrix (XM)∗ = M∗X∗ is easy to compute. The matrix M∗ has all its
coefficients equal to κ(G). The matrix X∗ equals diag((Qn x )/x ,...,(Qn x )/x ).
i=1 i 1 i=1 i n
The group Pic0(XΛ ) has order equal to the greatest common divisors of the coefficients
G
of (XM)∗. This integer is computed as
Yn Yn (Qn x )
|Pic0(XΛ )| = κ(G)gcd(( x )/x ,...,( x )/x ) = κ(G) i=1 i .
G i 1 i n
‘
i=1 i=1
(cid:3)
Remark 2.6 When |Pic0(XΛ )| = 1, we find that G is a tree and ‘ = Qn x . Then
G i=1 i
XΛ = Λ , and g(XΛ ) = g(R). It turns out that g(R) is well-understood already:
G R G
when ‘ = Qn x , the sequence ‘/x ,...,‘/x , is strongly flat, and g(R) = 1 + ‘(n −
i=1 i 1 n
1)−Pn ‘/x (see [23], 3.2.2 (b), or the original reference [22]). Proposition 2.4 lets us
i=1 i
interpret 1+‘(n−1)−Pn ‘/x as a g-number even when ‘ < Qn x ; it is the g-number
i=1 i i=1 i
of XΛ when G is a tree.
G
Corollary 2.7. Let Λ ⊆ Λ . Let x be a positive integer and consider xΛ ⊆ Λ.
R
a) g(xΛ)−1 = x(g(Λ)−1)+(x−1)(Pn r ).
i=1 i
b) Λ has a canonical vector if and only if xΛ has a canonical vector.
Ournextcorollaryshowsthatasearchforacanonicalvector for Λ ⊆ Λ canbe reduced
R
to a similar search for a lattice in Λ .
Jn
˜ ˜
Corollary 2.8. Let R > 0 and set R := diag(r ,...,r ). Let Λ ⊆ Λ , and RΛ ⊆ Λ .
1 n R Jn
a) g(R˜Λ) = g(Λ)+Pn (r −1).
i=1 i
˜
b) Λ has a canonical vector if and only if RΛ has a canonical vector.
2.9 We consider now the case of arithmetical graphs, and show that a canonical vector
exists in some cases. The notion of arithmetical graph was introduced in [15], and we
start by recalling its definition.
Let G be a finite unweighted connected multigraph on n vertices v ,...,v , without
1 n
loop edges. Let A denote its adjacency matrix. Consider a diagonal (n × n)-matrix
D = diag(δ ,...,δ ) with strictly positive diagonal entries, and a vector tR = (r ,...,r )
1 n 1 n
with R > 0 and gcd(r ,...,r ) = 1. Let M := D −A. The triple (G,M,R) is called an
1 n
arithmetical graph if MR = 0. When D = diag(d ,...,d ) with d the valency of v and
1 n i i
tR = (1,...,1), thematrixM isnothingbuttheLaplacianofG. LetΛ := Im(M) ⊆ Λ .
M R
We may denote g(Λ ) and Pic0(Λ ) simply by g(M) and Pic0(M), respectively.
M M
FROBENIUS NUMBER, RIEMANN-ROCH STRUCTURE, AND ZETA FUNCTIONS OF GRAPHS 7
The main geometric invariant of an arithmetical graph is the integer g (M) defined by
0
the expression
n n
X X
2g (M)−2 = r (δ −2) = r (d −2).
0 i i i i
i=1 i=1
That g (M) is always an integer is noted in [15], 3.6. When tR 6= (1,...,1), we do not
0
have examples of two arithmetical graphs (G ,M ,R) and (G ,M ,R) on n vertices such
1 1 1 2
that Im(M ) = Im(M ) and g (M ) 6= g (M ). In other words, we do not know whether
1 2 0 1 0 2
the integer g depends only on the lattice spanned by the columns of the matrix M.
0
In the Riemann-Roch theorem for the Laplacian of a graph, the canonical class is
represented by tK := (d − 2,...,d − 2), with Pn (d − 2) = 2β(G) − 2. For an
G 1 n i=1 i
arithmetical graph (G,M,R), a natural analogue to consider is
K := t(δ −2,...,δ −2),
1 n
with deg(K) = 2g (M)−2. Note that by adding together the columns of the matrix M
0
we obtain the vector t(δ −d ,...,δ −d ), showing that t(d −2,...,d −2) is a vector
1 1 n n 1 n
equivalent to K in Pic(Λ ).
M
Proposition 2.10. Let (G,M,R) be an arithmetical graph, with Λ := Im(M) ⊆ Λ .
M R
Let K := t(δ −2,...,δ −2). Then
1 n
(a) g(Λ ) ≤ g (M).
M 0
(b) Let D ∈ Zn with deg(D) = g (M)−1. Then (cid:15) (D) = (cid:15) (K −D). In particular,
0 ΛM ΛM
K is a canonical vector for Λ if g(Λ ) = g (M).
M M 0
Proof. We prove this proposition by first interpreting the matrix M as the intersection
matrix associated with the reduction of a curve, and then by applying the Riemann-Roch
Theorem for curves.
Given M and R, there exist a complete discrete valuation ring O (with field of frac-
F
tions F and algebraically closed residue field k of characteristic 0), and a smooth proper
geometrically connected curve X/F of (geometric) genus p (X) with a regular model
g
X/O satisfying the following properties (see [26], 4.3). The special fiber X /k of X/O
F k F
is the union of smooth irreducible curves C , i = 1,...,n (called the components of X ).
i k
Each curve C /k has genus 0 and multiplicity r . Denote by (C · C ) the intersection
i i i j
number of the components C and C on X. The matrix ((C · C )) is called the
i j i j 1≤i,j≤n
intersection matrix of X /k, and is equal to M. These conditions, on the intersection
k
matrix and on the genus of the components of X , imply that p (X) = g (M).
k g 0
Our reference for the facts recalled below is [14], 9.1. Since X is regular, the natural
inclusion X → X induces a surjective restriction map res : Pic(X) → Pic(X). Recall
that Pic(M) := Zn/Im(M). In keeping with the geometric notation, we write Zn as
Div(M) = ⊕n ZC , and call an element of Div(M) a divisor. Two divisor D and E are
i=1 i
said to be equivalent if [D] = [E]. We also have a natural homomorphism ρ : Pic(X) →
Pic(M) defined as follows: L ∈ Pic(X) is mapped to the class in Pic(M) of the divisor
deg(L )C + ··· + deg(L )C . There exists in Pic(X) an element K , called the
|C1 1 |Cn n X/OF
canonical bundle, with the following properties:
(i) ρ(K ) = [K], and
X/OF
(ii) res(K ) = K is the canonical bundle in Pic(X).
X/OF X/F
For each i, we can choose a closed point P of X whose closure P in X intersects X /k
i i k
only in C ; more precisely, we require the following: (P · C ) = 1, and (P · C ) = 0 if
i i i i j
j 6= i. This shows that the map ρ is surjective. Indeed, let D := Pn a C be a divisor
i=1 i i
of Div(M). Consider the line bundle L := O (Pn a P ). Then ρ(L) = [D]. Moreover,
X i=1 i i
FROBENIUS NUMBER, RIEMANN-ROCH STRUCTURE, AND ZETA FUNCTIONS OF GRAPHS 8
the degree of the divisor res(L) := O (Pn a P ) in Pic(X) is equal to the degree of [D]
X i=1 i i
in Pic(M).
(a) Let D := Pn a C be a divisor of Div(M) of degree r ≥ g (M). Consider the
i=1 i i 0
divisorD0 := Pn a P inDiv(X). Sincer ≥ g (M) = p (X),theRiemann-Rochtheorem
i=1 i i 0 g
on curves ([17], IX.4.1) shows that the vector space H0(X,D0) has positive dimension,
which implies that we can find a divisor E0 = Ps b Q ∈ Div(X) with b ≥ 0 for
j=1 j j j
all j and such that D0 and E0 are linearly equivalent. Consider the line bundle L0 :=
O (Ps b Q ). Then L0 ⊗L−1 is trivial on the generic fiber X. Hence, ρ(L) = ρ(L0) in
X j=1 j j
Pic(M). It is easy to verify that ρ(L0) is represented in Div(M) by an effective divisor.
By construction, ρ(L) = [D] and, thus, D is equivalent to an effective divisor. It follows
that g(Λ ) ≤ g (M).
M 0
To prove (b), it is sufficient to show that if D = Pn a C is an effective divisor of
i=1 i i
degree g (M)−1 in Div(M), then K−D is equivalent to an effective divisor. Consider the
0
divisor D0 := Pa P in Div(X). Let K0 ∈ Div(X) denote a canonical divisor for X/F.
i i
Since g (M) = p (X), the Riemann-Roch theorem for curves implies that either both D0
0 g
and K0 −D0 are linearly equivalent to an effective divisor on X, or neither is. Since D0
is clearly effective in Div(X) because D is, K0 − D0 is equivalent to an effective divisor
E0 = Ps b Q ∈ Div(X). Consider then the line bundle K ⊗O (Pn a P )−1 ⊗
j=1 j j X/OF X i=1 i i
O (Ps b Q )−1 in Pic(X), which is trivial on the generic fiber X by construction. It
X j=1 j j
follows that ρ(K ⊗O (Pn a P )−1) is equivalent to ρ(O (Ps b Q )) in Pic(M).
X/OF X i=1 i i X j=1 j j
By construction, ρ(K ⊗ O (Pn a P )−1) = [K − D], and ρ(O (Ps b Q )) is
X/OF X i=1 i i X j=1 j j
effective. (cid:3)
It would be of interest to find a completely combinatorial proof of this proposition. It
is also natural to wonder whether M has a canonical vector when g(M) < g (M). We
0
give below some examples of arithmetical graphs, and of the inequalities g(R) ≤ g(Λ ) ≤
M
g (M).
0
Example 2.11 Let R = (r ,...,r ) be an integer vector with positive entries such that
1 n
gcd(r ,...,r ) = 1. Using the incidence matrix of any connected graph H on n vertices,
1 n
we construct an arithmetical graph of the form (G,M,R), producing in this way for most
vectors R many non-isomorphic arithmetical graphs having the same vector R.
Indeed, given a graph H, Chung and Langlands introduced a Laplacian with vertex
weights in [7]. When, for all i, the weight of the vertex v is the square of a positive integer
i
r , then the Laplacian L introduced in [7], (1), p. 317, is the matrix of an arithmetical
i
graph with vector tR = (r ,...,r ). We recall this construction here. Let B denote the
1 n
incidence matrix of H, with n rows and m columns. The Laplacian of H is equal to
B(tB). Let B denote the following (n×m)-matrix. Say a row of the transpose tB has
R
+1 in its i-th entry and −1 in its j-th entry, then the matrix tB has +r in its i-th entry
R j
and −r in its j-th entry. By construction, (tB )R = 0. We let M := B (tB ), and
i R R R
denote by (G,M,R) the associated arithmetical graph.
Let r := Pn r2. When the initial graph H is the complete graph on n vertices, the
i=1 i
associated arithmetical graph (G,M,R) has M = rI − R(tR). The group Pic0(Λ ) is
n M
isomorphic to (Z/rZ)n−2 ([15], 1.10). When R = t(1,1,2), g(M) = g (M) = 4, and for
0
R = t(1,1,3), g(M) = 6 and g (M) = 9. In the latter case, t(5,5,0) is a canonical vector.
0
The simplest example of this construction is with the tree H on two vertices and the
vector tR = (r,s), gcd(r,s) = 1. Then the associated (G,M,R) has
(cid:18) (cid:19)
s2 −rs
M = .
−rs r2
FROBENIUS NUMBER, RIEMANN-ROCH STRUCTURE, AND ZETA FUNCTIONS OF GRAPHS 9
In this case, |Pic0(Λ )| = 1, so g(M) = g(r,s) = (r −1)(s−1). An easy computation
M
shows that g (M) = 1sr(r+s)−(r+s)+1.
0 2
Remark 2.12 When (G,M,R) is an arithmetical graph, let us denote by β(G) := m−
n+1 the first Betti number of the graph G. It is noted in [15], 4.7, that g (M) ≥ β(G). It
0
isnottrueingeneralthatg(M) ≥ β(G). Intheaboveexamplewhenn = 2, β(G) = rs−1,
while g(Λ ) = (r−1)(s−1).
M
Example 2.13 Alllatticesofrank1inZ2 haveacanonicalvector. Indeed, lettR := (r,s)
with gcd(r,s) = 1. The lattice Λ is generated in Z2 by the vector t(s,−r). We have
R
g(Λ ) = g(r,s) = (r−1)(s−1). Note that the lattice Λ is in fact the lattice associated
R R
with an arithmetical graph (G,M,R), namely, the arithmetical graph with M as in 2.11.
The only sublattices of Λ are of the form xΛ . Then g(xΛ ) = xrs − r − s + 1 and
R R R
K := (xs−2,xr−2) is a canonical vector for the lattice xΛ (2.7).
R
Example 2.14 (a) Given coprime integers 2 ≤ a < b, there exists an arithmetical graph
(G,M,R) such that |Pic0(M)| = 1 and g (M) = g(M) = g(R) = g(a,b).
0
Indeed, define positive integers s ,s ,s , and s by the equality 2ab = (s + s )a +
1 2 3 4 1 2
(s +s )b = a+(b−1)a+b+(a−1)b. We consider then an arithmetical tree (G,M,R)
3 4
with a single node v of multiplicity ab, and valency 4. Attached to v are four terminal
chains. The vertices linked to v have multiplicities s a, s a, s b, and s b, respectively.
1 2 3 4
The terminal vertices on the chains have multiplicities a, a, b, and b. Each terminal chain
is obtained using Euclid’s algorithm on the pairs (ab,s a) and (ab,s b), as in [15], 4.2.
i i
Since the terminal multiplicity of a chain divides each multiplicity on the chain, we find
that g(R) = g(a,b). A formula for the order of |Pic0(M)| when G is a tree is given
in [15], 2.5, and can be used to show that |Pic0(M)| = 1. Thus 1.3 (a) implies that
g(M) = g(R) = g(a,b). An easy computation shows that g (M) = g(a,b).
0
(b) Given coprime integers 2 ≤ a < b < c with gcd(a,c) = 1, it is often possible to find
an arithmetical graph (G,M,R) such that |Pic0(M)| = 1 and g(M) = g(R) = g(a,b,c).
In the example below, such an arithmetical graph has g (M) = 1+ 1(abc−a−b−c); in
0 2
particular, it is possible to find many instances where g(M) < g (M).
0
Indeed, suppose that we found positive integers s , s , and s , such that abc = s a+
1 2 3 1
s b+s c and gcd(abc,s a)/a = gcd(abc,s b)/b = gcd(abc,s c)/b = 1. For instance, since
2 3 1 2 3
gcd(a,c) = 1, find 0 < x < a such that a | c+bx. If gcd(x,c) = 1, take s = 1, s = x,
1 2
and s c = abc−a−bx > 0.
3
Consider the arithmetical tree (G,M,R) with a single node v of multiplicity abc and
valency 3 constructed as follows. Attached to v are three terminal chains. The vertices
linked to v have multiplicities s a, s b, and s c, respectively. The terminal vertices on
1 2 3
the chains have multiplicities a, b, and c. Each terminal chain is obtained using Euclid’s
algorithm on the pairs (abc,s a), (abc,s b), and (abc,s c), as in [15], 4.2. Since the
1 2 3
terminal multiplicity of a chain divides each multiplicity on the chain, we find that g(R) =
g(a,b,c). The formula for the order of |Pic0(M)| when G is a tree given in [15], 2.5, can be
appliedandshowsthat|Pic0(M)| = 1. Thus1.3(a)impliesthatg(M) = g(R) = g(a,b,c).
Recall ([4], page 20) that
1 p
g(a,b,c) ≤ 1+ ( abc(a+b+c)−a−b−c).
2
In particular, in this example, the difference g (M)−g(M) can be arbitrarily large.
0
Thepreviousexamplesshowhowtoconstructarithmeticaltreeswhereg(R) = g(Λ ) =
M
g (M) and where g(Λ ) < g (M). The integer g(Λ ) is in general hard to compute on
0 M 0 M
examples, and we do not know of large classes of arithmetical graphs with g(M) = g (M),
0
FROBENIUS NUMBER, RIEMANN-ROCH STRUCTURE, AND ZETA FUNCTIONS OF GRAPHS 10
except for the ones obtained using blow-ups. Recall that when tR = (1,...,1) (i.e., for
‘usual’ graphs), we have g (M) = g(M) = β(G). Given any arithmetical graph (G,M,R)
0
on n vertices, there is a construction called ‘blow-up’ which produces a new arithmetical
graph (G0,M0,R0) on n+1 vertices, and this new arithmetical graph is not a ‘usual’ graph
(2.17). Thus, starting with a graph G with g(M) = g (M), we can construct infinitely
0
many associated arithmetical graphs with g = g .
0
2.15 The following construction starts with a lattice Λ of rank n−1 having a canonical
vector K and creates new lattices of rank n also having canonical vectors. Fix n > 0
and a vector R > 0. Let Λ ⊆ Λ be a sublattice of rank n − 1 with g-number g.
R
Consider the natural inclusion ϕ : Zn → Zn+1, sending the basis vector e of Zn to the
i
basis vector e of Zn+1. Fix x := Pn x r > 0 and an integer s > 0 coprime to x, and
i i=1 i i
consider the vectors Q := t(x ,...,x ) ∈ Zn and (Q,−s) := t(x ,...,x ,−s) ∈ Zn+1.
1 n 1 n
Let R0 := (sR,x) ∈ Zn+1. Note that gcd(sr ,...,sr ,x) = 1. Let Λ(Q,s) denote the
1 n
lattice of Zn+1 generated by ϕ(Λ) and (Q,−s). It is easy to check that Λ(Q,s) ⊆ Λ .
R0
Proposition 2.16. Let Λ, Q, and s be as above. Then
a) The sequence 0 → Pic0(Λ) → Pic0(Λ(Q,s)) → Z/sZ is exact.
b) g(Λ(Q,s))−1 = s(g(Λ)−1)+(s−1)x.
c) If K is a canonical vector for Λ, then (K,2(s−1)) is a canonical vector for Λ(Q,s).
Proof. (a) We clearly have an exact sequence 0 → Pic(Λ) → Pic(Λ(Q,s)) → Z/sZ → 0.
Since taking the torsion subgroup is a left exact functor, (a) follows.
(b)LetD = t(d ,...,d ) ∈ Zn+1. ThevectorD isΛ(Q,s)-equivalenttoavectorwith
1 n+1
last coefficient in [0,s−1]. Without loss of generality, we may assume that 0 ≤ d ≤
n+1
s−1. Assume that deg (D) > s(g(Λ)−1)+(s−1)x. Then deg (d ,...,d ) > (g(Λ)−1),
R0 R 1 n
and t(d ,...,d ) is Λ-equivalent to an effective. Hence, D is Λ(Q,s)-equivalent to an
1 n
effective, and g(Λ(Q,s))−1 ≤ s(g(Λ)−1)+(s−1)x.
Assume now that deg (D) = s(g(Λ)−1)+(s−1)x. If d < s−1, then the argument
R0 n+1
above shows that D is Λ(Q,s)-equivalent to an effective. When d = s−1, then D is
n+1
Λ(Q,s)-equivalent to an effective if and only if t(d ,...,d ) is Λ-equivalent to an effective.
1 n
Hence, g(Λ(Q,s))−1 = s(g(Λ)−1)+(s−1)x.
(c) Note first that 2g(Λ(Q,s)) − 2 = 2s(g(Λ) − 1) + 2(s − 1)x = deg (K,2(s − 1)).
R0
We showed in (b) that the only vectors of degree deg (D) = g(Λ(Q,s)) − 1 which are
R0
not Λ(Q,s)-equivalent to an effective are the vectors D = (D0,s − 1), with D0 having
deg (D0) = g(Λ)−1 and not Λ-equivalent to an effective. It is clear that D → (K,2(s−
R
1))−D permutes these vectors since D0 → K −D0 is a permutation of the set of vectors
that are not Λ-equivalent to an effective. (cid:3)
Remark 2.17 Suppose that Λ = Im(M) is the lattice associated with an arithmetical
graph (G,M,R). Then, if Q ≥ 0, it may happen that Λ(Q,1) is also associated with an
arithmetical graph, called a blow-up (G0,M0,R0) of (G,M,R), with
(cid:18) (cid:19)
M +Q(tQ) −Q
M0 := .
−tQ 1
Indeed, for (G0,M0,R0) to be an arithmetical graph, M + Q(tQ) needs non-positive co-
efficients off the diagonal. For instance, if the coefficient m of M is negative for some
ij
i < j, we can obtain new arithmetical graph using the vector Q having only two non-zero
coefficients, a 1 in position i and a 1 in position j. It is easy to check that if M0 is a
blow-up of M, then g (M) = g (M0).
0 0