Realization of groups with pairing as Jacobians of finite graphs
Louis Gaudet, David Jensen, Dhruv Ranganathan, Nicholas Wawrykow, Theodore Weisman
aa r X i v : . [ m a t h . C O ] S e p REALIZATION OF GROUPS WITH PAIRING AS JACOBIANS OF FINITEGRAPHS
LOUIS GAUDET, DAVID JENSEN, DHRUV RANGANATHAN, NICHOLAS WAWRYKOW, ANDTHEODORE WEISMANA
BSTRACT . We study which groups with pairing can occur as the Jacobian of a finite graph.We provide explicit constructions of graphs whose Jacobian realizes a large fraction of oddgroups with a given pairing. Conditional on the generalized Riemann hypothesis, theseconstructions yield all groups with pairing of odd order, and unconditionally, they yieldall groups with pairing whose prime factors are sufficiently large. For groups with pairingof even order, we provide a partial answer to this question, for a certain restricted class ofpairings. Finally, we explore which finite abelian groups occur as the Jacobian of a simplegraph. There exist infinite families of finite abelian groups that do not occur as the Jacobiansof simple graphs.
1. I
NTRODUCTION
Given a finite graph G , there is naturally associated group Jac ( G ) , the Jacobian of G . Thegroup Γ = Jac ( G ) comes with a symmetric, bilinear, non-degenerate pairing [10, 14], h· , ·i : Γ × Γ → Q / Z , known as the monodromy pairing . Groups with such a pairing will be referred to simply as groups with pairing . Clancy, Leake, and Payne [6] observed that the Jacobian of a randomlygenerated graph is cyclic with probability close to . This probability agrees with thewell-known Cohen–Lenstra heuristics, which predict that a finite abelian group Γ shouldoccur with probability proportional to | Aut ( Γ ) | . However, other classes of groups violatethese heuristics. This is because the Jacobian of a graph should really be thought of asa group, together with a duality pairing. In loc.cit., it is conjectured that a group withpairing ( Γ, h· , ·i ) should occur with probability proportional to | Γ || Aut ( Γ, h· , ·i ) | . This is furthersuggested by the empirical evidence of [5] and proven in [16].Given a finite abelian group with pairing Γ , the probability that a random graph hasJacobian isomorphic to Γ is zero [16], so it is possible that some groups with pairing do notoccur at all. In the present text, we investigate precisely which finite abelian groups withpairing can occur as the Jacobian of a finite graph. Our main result is the following. Theorem 1.
Let Γ be a finite abelian group with pairing. There exists a finite set of primes P ⊂ Z such that, if | Γ | is not divisible by any p ∈ P , then there exists a graph G such that Γ ∼ = Jac ( G ) as groups with pairing. It is our expectation that the set of primes P appearing in Theorem 1 consists of onlythe prime 2. We have the following result, conditional on the generalized Riemann hy-pothesis [8]. Date : September 19, 2017. heorem 2 (Conditional on GRH) . Let Γ be a finite abelian group with pairing of odd order.Then there exists a graph G such that Γ ∼ = Jac ( G ) as groups with pairing. Remark 3.
The above results are related to the following purely number theoretic question.
Given a prime p , does there exist a prime q < 2 √ p , with q ≡ mod , such that q is a quadraticnon-residue modulo p ? Numerical evidence suggests that this condition should be satisfiedfor all sufficiently large primes p .An interesting variation on the question considered here was studied by Bosch andLorenzini in [4, Proposition 5.2]. They consider the representation of groups with pair-ing arising from arithmetical graphs . While the strategy of our proof bears some similaritiesto that found in loc. cit., the presence of arithmetical structure simplifies the classifica-tion problem. Indeed, as shown in [4, Example 5.4], in the case of arithmetical graphs onecan take the underlying graph to be a tree. Our setting is motivated by considerations intropical geometry and the graph theoretic Abel–Jacobi theory of Baker and Norine.Jacobians of wedge-sums of graphs decompose canonically as the orthogonal direct sumof the Jacobians of their components. A structure theorem for groups with pairing there-fore allows us to focus primarily on the case where Γ is cyclic. When Γ is a -group, how-ever, this structure result is more complicated. There are non-exceptional natural pairingson the group Z /2 r Z , and we find graphs which realize these groups with pairings. Thereare, in addition, exceptional families of pairings on the group ( Z /2 r Z ) that do not de-compose as the orthogonal direct sum of cyclic groups with pairing. We refer to Section 2for background regarding pairings on -groups. Theorem 4.
Let Γ ∼ = ( Z /2 r Z , h· , ·i ) be a cyclic -group with non-exceptional pairing h· , ·i . Thenthere exists a graph G such that Γ ∼ = Jac ( G ) as groups with pairing. We discuss groups with exceptional pairings in further detail in Section 4.2.If we forget the structure of the pairing on Γ , it is elementary to observe that everyfinite abelian group Γ occurs as the Jacobian of a multigraph G . Naively, however, theconstruction often necessitates the use of graphs with multiple edges. Since the Erd ˝os–R´enyi random graphs studied in [5, 6, 16] are always simple, we find it natural to ask thefollowing. Question.
Which finite abelian groups (without a specified pairing) occur as the Jacobianof a simple graph?We find that there are infinite families of finite groups that do not occur as the Jacobiansof simple graphs.
Theorem 5.
For any k > , there exists no simple graph G such that Jac ( G ) ∼ = ( Z /2 Z ) k . More generally, we have the following result for groups with a large number of Z /2 Z invariant factors. Theorem 6.
Let H be a finite abelian group. Then there exists a natural number k H depending on H , such that for all k > k H , there does not exist a simple graph G with Jac ( G ) ∼ = ( Z /2 Z ) k × H. cknowledgements. This project was completed as part of the 2014 Summer Undergrad-uate Mathematics Research at Yale (SUMRY) program, where the second and third authorswere supported as mentors and the first, fourth, and fifth authors were supported as partic-ipants. It is a pleasure to thank all involved in the program for creating a vibrant researchcommunity. We benefited from conversations with Dan Corey, Andrew Deveau, JennaKainic, Nathan Kaplan, Susie Kimport, Dan Mitropolsky, and Anup Rao. We thank SamPayne for suggesting the problem. We are also especially grateful to Paul Pollack, whoseideas significantly strengthened the results of this paper. Finally, we thank the referees fortheir careful reading and insightful comments.The authors were supported by NSF grant CAREER DMS-1149054 (PI: Sam Payne).2. B
ACKGROUND
Jacobians of graphs.
We briefly recall the basics of divisor theory on graphs. Werefer to [2] for further details. In this paper a graph will mean a finite connected graph,possibly with multiple edges, but without loops at vertices. A simple graph is a graphwithout multiple edges. A divisor on a graph is an integral linear combination of vertices,and we write a divisor as D = X v ∈ V ( G ) D ( v ) v, where each D ( v ) is an integer. The degree of a divisor D is deg ( D ) = X v ∈ V ( G ) D ( v ) . It is common to think of a divisor as a configuration of “chips” and “anti-chips” on thevertices of the graph, so that the degree is just the total number of chips.Let M ( G ) := Hom ( V ( G ) , Z ) be the group of integer-valued functions on the vertices of G . For f ∈ M ( G ) , we define ord v ( f ) := X e = vw edge containing v ( f ( v ) − f ( w )) , and div ( f ) := X v ∈ V ( G ) ord v ( f ) v. Divisors that arise as div ( f ) for a function f ∈ M ( G ) are referred to as principal . We say thattwo divisors D and D are equivalent , and write D ∼ D , if their difference is principal.Equivalence of divisors is related to the well-known “chip-firing game” on graphs,which can be described as follows. Given a divisor D and a vertex v , the chip-firing move centered at v corresponds to the vertex v giving one chip to each of its neighbors. That is,the vertex v loses a number of chips equal to its valence, and each neighbor gains exactly chip. Two divisors are equivalent if one can be obtained from the other by a sequence ofchip-firing moves.Note that the degree of a divisor is invariant under equivalence. The Jacobian
Jac ( G ) isthe group of equivalence classes of divisors of degree zero. The Jacobian of a connectedgraph is always a finite group, with order equal to the number of spanning trees in G ,see [3]. or the most part, we will not need any deep structural results about the Jacobians ofgraphs. The following result, however, will greatly simplify one of our proofs in the latersections. Theorem 7. [7, Theorem 2]
Let G be a planar graph and let G ⋆ be a planar dual of G . Then, theJacobian of G and G ⋆ are isomorphic as groups. The Jacobian of a graph comes equipped with a bilinear pairing, known as the mon-odromy pairing , defined as follows. Given two divisors D , D ∈ Jac ( G ) , first find an in-teger m such that mD is principal – that is, there exists a function f ∈ M ( G ) such that div ( f ) = mD . Then we define h D , D i = X v ∈ V ( G ) D ( v ) f ( v ) . It is of course not immediately clear that the pairing above is non-degenerate. A proofmay be found in [14, Theorem 3.4].
Remark 8.
Note that the isomorphism of Jacobians of planar dual graphs does not in gen-eral preserve the pairings. See for instance Corollary 16.2.2.
Reduced divisors and Dhar’s burning algorithm.
Given a divisor D and a vertex v ,we say that D is v -reduced if(1) D ( v ) > for all vertices v = v , and(2) every non-empty set A ⊆ V ( G ) r { v } contains a vertex v such that outdeg A ( v ) >D ( v ) .By [2, Proposition 3.1], every divisor is equivalent to a unique v -reduced divisor.There is a simple algorithm for determining whether a given divisor satisfying (1) aboveis v -reduced, known as Dhar’s burning algorithm . For v = v , imagine that there are D ( v ) buckets of water at v . Now, light a fire at v . The fire consumes the graph, burning anedge if one of its endpoints is burnt, and burning a vertex v if the number of burnt edgesadjacent to v is greater than D ( v ) (that is, there is not enough water to fight the fire). Thedivisor D is v -reduced if and only if the fire consumes the whole graph. For a detailedaccount of this algorithm, we refer to [3, Section 5.1] and [9].2.3. Jacobians of wedge sums of graphs.
Given two graphs with distinguished vertices ( G , v ) and ( G , v ) , the wedge sum is the graph formed by identifying v and v . Wesuppress the dependency on the choice of distinguished vertices in what follows, as thechoice will not matter, denoting the wedge sum as G ∨ G . A key tool in our proof isthe fact that the Jacobian of a wedge sum of graphs is the orthogonal direct sum of theJacobians. Proposition 9.
Let G , G be graphs. Then Jac ( G ∨ G ) ∼ = Jac ( G ) ⊕ Jac ( G ) , where ⊕ denotes the orthogonal direct sum of finite abelian groups with pairing.Proof. This follows from the fact that any piecewise linear function on G corresponds to apiecewise linear function on G i by restriction, and conversely any function on G i can beextended to a function on G by giving it a constant value on G r G i . (cid:3) ∨ G = G ∨ G F IGURE
1. The wedge sum operation on graphs. In this case,
Jac ( G ) ∼ = Z /3 Z , Jac ( G ) ∼ = Z /4 Z , and Jac ( G ∨ G ) ∼ = Z /12 Z .2.4. Structure results for groups with pairing.
Our arguments will rely heavily on theclassification of finite abelian groups with pairing from [12, 15]. A first step in this classifi-cation is the following.
Lemma 10.
Let Γ be a group with pairing h· , ·i , and suppose that there exist subgroups Γ , Γ ⊆ Γ such that Γ ∼ = Γ × Γ as groups. If the orders of Γ and Γ are relatively prime, then Γ is isomorphicto the orthogonal direct sum Γ ⊕ Γ . Lemma 10 reduces the classification of finite abelian groups with pairing to the classifi-cation of p -groups with pairing. In light of Proposition 9, this lemma allows us to focus onconstructing graphs whose Jacobian is a given p -group with pairing.If p is an odd prime, then there are precisely two isomorphism classes of pairings on Z /p r Z , for r > . More precisely, every nondegenerate pairing on Z /p r Z is of the form h x, y i a = axyp r for some integer a not divisible by p . Two such pairings h· , ·i a , h· , ·i b are isomorphic if andonly if the Legendre symbols of a and b are equal. We will refer to these two pairings asthe residue and nonresidue pairings. The following is a fundamental result for groups withpairing. Theorem 11. If p is an odd prime, then every finite abelian p -group with pairing decomposes asan orthogonal direct sum of cyclic groups with pairing. When p = , the situation is somewhat more intricate. Up to isomorphism, there are 4distinct isomorphism classes of pairings on Z /2 r Z , which we refer to as the non-exceptionalpairings . These are given below. A r ∼ = ( Z /2 r Z , h· , ·i ) , r > ; h x, y i = xy2 r B r ∼ = ( Z /2 r Z , h· , ·i ) , r > ; h x, y i = − xy2 r C r ∼ = ( Z /2 r Z , h· , ·i ) , r > ; h x, y i = r D r ∼ = ( Z /2 r Z , h· , ·i ) , r > ; h x, y i = − r . In addition, on ( Z /2 r Z ) there are two isomorphism classes of pairings that do not de-compose as an orthogonal direct sum of cyclic groups with pairing. We refer to these asthe exceptional pairings : r ∼ = (( Z /2 r Z ) , h· , ·i ) , r > ; h e i , e j i = (cid:14)
0, i = j r , otherwise F r ∼ = (( Z /2 r Z ) , h· , ·i ) , r > ; h e i , e j i = (cid:14) r − , i = j r , otherwise , where e i and e j are generators for ( Z /2 r Z ) .We note the following two results of Miranda [12]. Lemma 12.
Let Γ be a finite abelian group of order r , with pairing h· , ·i . If h x, x i = a2 r forsome x ∈ Γ and odd positive integer a , then Γ is cyclic generated by x . Furthermore, for some c ∈ { ± ± } , with c ≡ a ( mod ) , there is an isomorphism of groups φ : Γ → Z /2 r Z such that h x, y i = cφ ( x ) φ ( y ) r . Theorem 13.
The groups A r , B r , C r , D r , E r , F r generate all 2-groups with pairing underorthogonal direct sum.
3. O
DD GROUPS WITH PAIRING
In this section, we investigate which groups with pairing of odd order occur as theJacobian of a graph. The decomposition of the Jacobain of a wedge sum as the orthogonalsum of the Jacobians of its components reduces our goal to the following.
Problem.
Given a pairing h· , ·i on the group Z /p r Z with p odd, find a graph G such that Jac ( G ) is isomorphic to Z /p r Z , such that h· , ·i is induced by the monodromy pairing.When p = , which we consider in Section 4, we must also consider the non-decomposablepairings on Z /2 r Z × Z /2 r Z .3.1. Subdivided Banana Graphs.
We begin with the following construction.
Construction 1.
Let s = ( s , . . . , s m ) be a tuple of positive integers. Let B m denote theso-called “banana graph”, which has two vertices and m edges between them. Constructthe s -subdivided banana graph from B m by subdividing the i th edge s i − times. We denotethis graph by B s , see Figure 2. v wwv F IGURE
2. The -banana graph and the subdivided banana B ( ) . Proposition 14.
Fix a prime p and an integer r . Let s = ( s , . . . , s m ) be a tuple of positiveintegers such that m X i = Q mj = s j s i = p r and gcd ( s i , p ) = for all i . Then Jac ( B s ) ∼ = ( Z /p r Z , h· , ·i ) , here h· , ·i is the pairing on Z /p r Z given by h x, y i = ( Q mi = s i ) xyp r . Proof.
We first show that | Jac ( B s ) | = p r . Every spanning tree of B s is obtained by deletingone edge each from all but one of the subdivided edges of B m . It follows that the numberof spanning tees of B s is m X i = Q mj = s j s i = p r . We now show that
Jac ( B s ) is cyclic by exhibiting a generator. Let v and w be the twovertices of B s of valence m pictured in Figure 2, and consider the divisor D = v − w . Notethat the order of D must be a power of p , and let t r be the smallest nonnegative integersuch that p t D is equivalent to . By definition, there exists a function f : V ( G ) → Z suchthat div ( f ) = p t D .Orient the graph so that the head of each edge points toward w , and for each edge e with head x and tail y , let b ( e ) = f ( x ) − f ( y ) . Since D ( v ) = for any v ∈ V ( G ) r { v, w } ,we must have b ( e ) = b ( e ) for any two edges in the same subdivided edge of B m , andwe may therefore write b i = b ( e ) for any edge e in the i th subdivided edge. Observethat b i s i = f ( w ) − f ( v ) for all i . As div ( f ) = p t D , we may conclude that P mi = b i = p t .Consequently, p t = m X i = f ( w ) − f ( v ) s i = ( f ( w ) − f ( v )) p r Q mi = s i . From this, we deduce m Y i = s i = p r − t ( f ( w ) − f ( v )) . Since gcd ( s i , p ) = for all i , this is impossible unless r = t , and thus the group is cyclic,generated by D .The monodromy pairing on Jac ( B s ) is fully determined by the value of h D, D i . Considera function f : V ( G ) → Z such that b i = Q mj = s j s i . We see that div ( f ) = p r D , and hence h D, D i = Q mi = s i p r . (cid:3) Remark 15.
We have recently become aware that Proposition 14 was proven earlier in [10,Section 2]. We nevertheless reprove it here, as the argument is simple and the bananagraph B s is central to our later constructions.The cycle graph C n and the banana graph B n are both special cases of the subdividedbanana. The following is an immediate corollary. Corollary 16.
For any prime p and integer r , Jac ( B p r ) ∼ = ( Z /p r Z , h· , ·i ) Jac ( C p r ) ∼ = ( Z /p r Z , h· , ·i − ) , where h· , ·i and h· , ·i − are the pairings on Z /p r Z given by h x, y i = xyp r h x, y i − = (− ) xyp r . .2. Results on quadratic residues.
Observe that the monodoromy pairing on
Jac ( B p r ) isthe residue pairing on Z /p r Z . To achieve the nonresidue pairing, we will use the subdi-vided banana graph B s for an appropriate choice of s . Our approach will rely on quadraticreciprocity, and it will be necessary to consider the cases p ≡ ( mod ) and p ≡ ( mod ) separately. Proposition 17.
For any sufficiently large prime p , there exists a prime quadratic nonresidue q ≡ ( mod ) , such that q is less than √ p .Proof. Let χ be the nontrivial character mod and χ the quadratic character mod p ,and let X be the group of Dirichlet characters generated by χ and χ . The group X hasconductor f = lcm (
4, p ) = and exponent dividing n = . Define the form χ = + χ χ − χ − χ . By [13, Theorem 1.4], there exists an odd prime q ≪ ( ) + ǫ f ǫ ≪ + such that χ ( q ) = . By construction, however, if χ ( q ) = then χ ( q ) = χ ( q ) = − . Itfollows that q is a quadratic nonresidue and q ≡ ( mod ) . (cid:3) We will also need the following proposition
Proposition 18.
For any sufficiently large prime p and integer r > 1 , there exist nonresidues q = mod , q = mod with q , q < 2 √ p r .Proof. As in the previous proof, let χ be the nontrivial character mod and χ the qua-dratic character mod p . To ask for a prime quadratic nonresidue q ≡ mod is to ask fora prime q such that χ ( q ) = χ ( q ) = − . Consider the abelian field extension K of Q givenby K = Q ( √ − √ α ) , where α = (− ) p − p. The extension K is degree with conductor . The characters χ and χ are quadratic,and thus we may apply [13, Theorem 1.7], to obtain an upper bound on the prime q , q ≪ + ǫ . Now for the mod case, we simply replace χ ( q ) = χ ( q ) = − above with the condi-tions χ ( q ) =
1, χ ( q ) = − and apply [13, Theorem 1.7] again. (cid:3) Proposition 19 (Conditional on GRH) . For any prime p > 10 , there exists a prime quadraticnonresidue q ≡ ( mod ) such that q < 2 √ p .Proof. Let α = (− ) p − p , and let K = Q ( √ − √ α ) . The degree of the extension K/ Q is 4,and the discriminant is ( ) . By [1, Theorem 5.1], by assuming GRH, that there exists aprime quadratic nonresidue q ≡ ( mod ) satisfying q < ( log ( ) + ) . The term on the right is smaller than √ p as long as p > 10 . (cid:3) iven a prime q that satisfies the bounds above, we will need to find a particular wayto write it as a sum of two positive integers, to ensure that s has the desired properties.Below, we check that such a decomposition exists, and that this decomposition providesthe properties we require. Lemma 20.
Let q be an odd prime, and let k be an integer such that (cid:16) kq (cid:17) = (cid:16) − (cid:17) . Then thereexists such that a ( q − a ) ≡ k ( mod q ) .Proof. Consider the set R q = (cid:12) ℓ ∈ F q : (cid:18) ℓq (cid:19) = (cid:18) − (cid:19) (cid:13) , and the map φ : F q → F q given by φ ( x ) = − x . The image of φ must be a subset of R q .For a fixed a , the polynomial x + a has at most two roots in F q . Since | R q | = q − , φ musttherefore surject onto R q . Hence, there exists an integer a such that φ ( a ) = k , and we have k ≡ − a ≡ a ( q − a ) ( mod q ) , as required. (cid:3) Lemma 21.
Let p be a sufficiently large prime with p ≡ ( mod ) , and let r be an integer. Thenthere exists a prime q , with (cid:16) qp r (cid:17) = − , and a positive integer a < q such that the quantity p r − a ( q − a ) q is a positive integer.Proof. By Proposition 18, there exists a nonresidue q with (cid:16) − (cid:17) = (cid:16) p r q (cid:17) , and q < p r . ByLemma 20, there exists a positive integer a < q such that p r ≡ a ( q − a ) ( mod q ) . Therefore p r − a ( q − a ) is positive and divisible by q . (cid:3) We now apply Lemma 21 to establish the existence of an s such that Jac ( B S ) ∼ = Z /p r Z with the nonresidue pairing. Proposition 22.
For any sufficiently large prime p and integer r , there exists s = { s , . . . s m } such that m X i = Q mj = s j s i = p r , gcd ( p, s i ) = for all i , and Q mi = s i is a nonresidue modulo p .Proof. First consider the case that p ≡ ( mod ) . Choose s = {
1, p r − } , and note that p r − ≡ − ( mod p r ) is a nonresidue modulo p r .In the case that p ≡ ( mod ) , let q, a be as in Lemma 21, and let s = a, s = q − a, s = p r − a ( q − a ) q . Since both a and q − a are smaller than p , they are relatively prime to p , and thereforethe product a ( q − a ) is relatively prime to p as well. Now, the quantity s s s is a non-residue mod p r iff (− )( a ( q − a )) q is a nonresidue mod p . Since p ≡ ( mod ) , − is aresidue modulo p r , and hence the numerator of this expression is also a residue. Therefore (cid:16) s s s p r (cid:17) = (cid:16) qp r (cid:17) = − , and the result follows. (cid:3) .3. Proof of Theorems 1 and 2.
Proof of Theorem 1.
By Corollary 16,
Jac ( B p r ) ∼ = Z /p r Z with the residue pairing. By Propo-sitions 14 and 22, for any sufficiently large prime p and integer r > , there exists an s suchthat Jac ( B s ) ∼ = Z /p r Z with the nonresidue pairing. By taking wedge sums of these graphs,we obtain all groups with pairing of odd order. (cid:3) Our proof of Theorem 2 is aided by the fact that in certain cases, we can explicitly con-struct an s satisfying the conditions required to achieve the nonresidue pairing: Proposition 23.
Let p be an odd prime, not equivalent to ( mod ) , and r > an integer. Thenthere exists an s such that m X i = Q mj = s j s i = p r , and Q mi = s i is a nonresidue modulo p .Proof. We consider the following three cases.(A) When p ≡ ( mod ) , as before, we may use s = {
1, p r − } .(B) When p ≡ ( mod ) , use s = {
1, 1, p r − } . Since p ≡ ( mod ) , the product s s s isa nonresidue modulo p iff is a nonresidue modulo p —which is the case when p ≡ ( mod ) .(C) When p ≡ ( mod ) , if p ≡ ( mod ) , we are in the first case above. Otherwise,we have p ≡ ( mod ) , and is a nonresidue modulo p . Choose s = {
1, 1, p r − } asbefore.The only remaining possibility after eliminating these three cases is p ≡ ( mod ) . (cid:3) Remark 24.
Proposition 23 shows that we could provide an unconditional proof of Theo-rem 2 if we could show that Proposition 19 holds for all primes p ≡ ( mod ) . In fact,computer search has verified that the proposition holds for all such primes smaller than . The code is available upon request of the authors. Proof of Theorem 2.
By Corollary 16,
Jac ( B p r ) ∼ = Z /p r Z with the residue pairing. By Propo-sitions 14 and 23, for any odd prime p not congruent to ( mod ) and integer r > , thereexists an s such that Jac ( B s ) ∼ = Z /p r Z with the nonresidue pairing. By Propositions 19 and22, if we assume GRH, then for any prime p > 10 and integer r > , there exists an s suchthat Jac ( B s ) ∼ = Z /p r Z with the nonresidue pairing. Finally, the computer search referencedin Remark 24 shows that, for all primes p ≡ ( mod ) , p < 10 , there exists an s suchthat Jac ( B s ) ∼ = Z /p r Z with the nonresidue pairing. Using the wedge sum construction, wemay obtain all groups with pairing of odd order, as desired. (cid:3)
4. 2-
GROUPS WITH PAIRING
We now turn to the task of constructing graphs G for which Jac ( G ) ∼ = (( Z /2 r Z ) k , h· , ·i ) for given positive integers r and k , and pairing h· , ·i . For each of the non-exceptional pair-ings on Z /2 r Z , we find a graph whose Jacobian is isomorphic to Z /2 r Z with the givenpairing. .1. Multicycle graphs.
In addition to the subdivided banana graphs of Section 3.1, wewill require one more construction.
Construction 2.
Let s = ( s , . . . , s m ) be a tuple of positive integers. Construct the s -multicycle graph C s on the vertices v , . . . , v m by introducing s i edges between v i and v i + (here i is taken mod m ), see Figure 3. v v v v F IGURE
3. The C ( ) multicycle graph.Note that the graphs B s and C s are planar duals of each other, and thus by Theorem 7, Jac ( B s ) ∼ = Jac ( C s ) as groups, but not necessarily as groups with pairing.We now show that all of the cyclic 2-groups with non-exceptional pairing are realizableas Jacobians of graphs. Theorem 25.
Let Γ ∼ = ( Z /2 r Z , h· , ·i ) . Then there exists a graph G such that Jac ( G ) ∼ = Γ .Proof. Observe that, by Corollary 16,
Jac ( B r ) ∼ = A r and Jac ( C r ) ∼ = B r . It remains tofind constructions for graphs providing the groups C r and D r .By Lemma 12, it suffices to find graphs G and G , with Jac ( G ) ∼ = Jac ( G ) ∼ = Z /2 r Z ,such that for some D ∈ Jac ( G ) and D ∈ Jac ( G ) , we have h D , D i = a2 r h D , D i = b2 r , where a ≡ ( mod ) and b ≡ − ( mod ) .We consider the cases for even and odd r separately. For odd r , let s = {
1, 2, r − } , andlet G = B s , G = C s . r −
23 2 r − F IGURE
4. The graphs B s and C s , for s = {
1, 2, r − } onsider a function f : V ( B s ) → Z , given by v ′ n − − j. If D = v − v , then div ( f ) = r D . It follows that h D , D i = f ( v ) r = r − r , asrequired.Now consider the function f : V ( C s ) → Z given by v
0, v
2, v If D = v − v , then div ( f ) = r D , so h D , D i = r , as desired.For even r , let s = {
1, 1, 1, r − } , and again let G = B s and G = C s . r −
13 2 r − F IGURE
5. The graphs B s and C s , for s = {
1, 1, 1, r − } For the banana graph, we see from Proposition 14 that
Jac ( B s ) is cyclic of order r , withpairing h x, y i = r − xy2 r . For the multicycle graph, consider a function f : V ( C S ) → Z , defined by f ( v i ) = i . If D = v − v , then div ( f ) = − r D , hence h D , D i = r , and the result follows. (cid:3) -groups with exceptional pairings. Each of the above constructions gives a graphwith cyclic Jacobian, giving four of the six generators for 2-groups with pairing. We havefew concrete results concerning the exceptional pairings. However, we make the followingobservation.
Proposition 26.
For any k > , there is no graph G such that Jac ( G ) ∼ = ( E ) k .Proof. This is a result of the characterization of graphs G with Jac ( G ) ∼ = ( Z /2 Z ) , givenbelow in Remark 31. Since the Jacobian of a cycle always gives rise to the group A , anysuch graph has Jacobian ( A ) . (cid:3) This result, combined with our failure to find any graph G that yields the group E r ,leads us to make the following conjecture: Conjecture 27.
For any k > , there is no graph G such that Jac ( G ) ∼ = ( E r ) k . e note, however, that there do exist examples of graphs G such that a subgroup H ⊂ Jac ( G ) (with the restricted pairing) is isomorphic to E r . For example, Jac ( B ) ∼ =( Z Z ) × Z /3 Z , and by inspection we can see that the -part with the restricted monodromypairing is isomorphic to E . F IGURE
6. The graph B .We have even fewer results regarding F r . We note that the complete graph K is a graphwith Jacobian isomorphic to F , but we were unable to find other examples of graphs thatprovide this pairing. 5. J ACOBIANS OF SIMPLE GRAPHS
In this section, we consider which groups without a specified pairing occur as Jacobiansof simple graphs. If a finite abelian group Γ does not have 2 as an invariant factor, then itis straightforward to construct a simple graph G such that Jac ( G ) ∼ = Γ , so this question isonly interesting for groups of the form ( Z /2 Z ) k × H .5.1. Preliminaries for proof of Theorem 5.
We first observe that any simple graph thathas spanning trees must have a third. To see this, consider the union of a spanning treewith a single edge not contained in the spanning tree. This union contains a cycle, andthe complement of any edge in this cycle is a spanning tree. Since the graph is simple,however, this cycle must contain at least three edges.Since the number of spanning trees is equal to the size of the Jacobian, there is no simplegraph G with Jac ( G ) ∼ = Z /2 Z .Many of our arguments focus on the case where the graph G is biconnected. Recall thata graph G is biconnected if for any vertex v ∈ V ( G ) , the induced subgraph on V ( G ) \ { v } is connected. In particular, if G is not biconnected, then by definition, there is a vertex v such that the induced subgraph on V ( G ) \ { v } is not connected. The graph G is thereforethe wedge sum of the connected components, which implies that Jac ( G ) splits as a directproduct of Jacobians. Definition 28.
Given a graph G , we write µ ( G ) for the maximum order of an element of Jac ( G ) ,and δ ( G ) for the maximum valency of a vertex in G . When the graph G is clear from context, wewill simply write δ and µ . Lemma 29.
For any biconnected graph G , δ ( G ) µ ( G ) . Furthermore, if δ ( G ) = µ ( G ) , then G must be the banana graph B µ .Proof. The statement is immediate if G consist of a single vertex, so we assume that G hasat least vertices. Let v be a vertex in V ( G ) with valency δ , and let w be a vertex adjacentto v . Consider the divisor D = v − w , and let m < δ be a positive integer. We applyDhar’s burning algorithm to check that mD is w -reduced. From the biconnectivity of G ,we deduce that there is a path from w to each of the neighbors of v that does not contain v . Thus, each of the neighbors of v is burned. By definition, val ( v ) > m , so it is burned s well. This means that mD cannot be equivalent to as is the unique reduced divisorequivalent to . It follows that D has order at least δ .In the case that δ = µ , we must have δD ∼ . Starting from δD , chip-fire v once to obtaina divisor E . Applying the burning algorithm and the biconnectivity condition once more,we see that v , as well as each of its neighbors, must be burned, so that E is w -reduced. E must therefore be the zero divisor, which is only possible if the multiplicity of the edge { v, w } is δ , i.e. G is a banana graph. (cid:3) Recall that the genus of a graph G is its first Betti number, given by g = | E ( G ) | − | V ( G ) | + . Corollary 30.
For any biconnected graph G with genus g and | V ( G ) | = n , n > − − Proof.
Let e be the total number of edges in G . We have an inequality = n X i = val ( v i ) n X i = δ = n · δ n · µ. Since e = g + n − , we see that − n · ( µ − ) . (cid:3) We are now ready to prove Theorem 5.
Proof of Theorem 5.
Let G be a simple graph with Jac ( G ) ∼ = ( Z /2 Z ) k . We may assume that G has no vertices of valence 1, because the graph obtained by contracting the edge adjacentto such a vertex has isomorphic Jacobian. If G is not biconnected, then G decomposes asa wedge sum, and Jac ( G ) decomposes as a direct sum of Jacobians, one of which mustbe isomorphic to ( Z /2 Z ) r for some positive integer r k . We may therefore assume that G is biconnected. By Lemma 29, it also has no vertices of valence 3 or greater. It followsthat G is a cycle. Since Jac ( C n ) ∼ = Z /n Z , we must have n = , which means G cannot besimple. (cid:3) Remark 31.
The proof of Theorem 5 also gives a complete characterization of graphs G with Jac ( G ) ∼ = ( Z /2 Z ) k . In general, we can always obtain such a graph by the followingprocedure. Start with a tree T , and choose a subset of k edges of T . Construct a new graph G from T by doubling each edge in this subset. See Figure 7.F IGURE
7. An example of a graph G with Jac ( G ) ∼ = ( Z /2 Z ) .2. Preliminaries: Proof of Theorem 6.
Our next goal is to generalize Theorem 5 tographs whose Jacobian is of the form ( Z /2 Z ) k × H . We begin with the following bound onthe genus of G . Proposition 32. [11, Proposition 5.2] If G is a graph of genus g and Jac ( G ) ∼ = ( Z /2 Z ) k × H ,then g > k . Applying Corollary 30 to this result shows that | V ( G ) | > − − We require the following result about lengths of paths in G . Lemma 33.
Let G be a biconnected graph, and suppose that there exists a path P with vertices { v , . . . , v ℓ } on G such that val ( v i ) = for all . Then Jac ( G ) contains an element oforder at least ℓ .Proof. Let m < ℓ , and consider D = v − v . As G is biconnected, there is a path from v to v m + that does not contain any of the vertices of P . Dhar’s burning algorithm shows that v m + − v is the v -reduced divisor equivalent to mD , and hence mD ≁ for m < ℓ . (cid:3) Our approach will now be to establish an upper bound on | V ( G ) | in terms of µ and | H | ,and then use this to obtain an upper bound on k . Proposition 34.
For any finite abelian group H , there exists an integer n H such that, for anybiconnected simple graph G with Jac ( G ) ∼ = ( Z /2 Z ) k × H , we have | V ( G ) | < n H .Proof. Let U = { u ∈ V ( G ) : val ( u ) > 2 } . We will first establish a bound on m = | U | , andthen bound | V ( G ) | in terms of m .Fix a vertex u ∈ U , and consider the set of divisors U = { u i − u | u i ∈ U } . For any D = D ∈ U , we claim that − = − is u -reduced. Since G is biconnected,there is a path from u to each of the neighbors of u that does not contain u . ApplyingDhar’s burning algorithm, we see that since val ( u ) > 2 , the entire graph will be burned.Therefore − is u -reduced, hence ≁ .We now define a map ϕ : Jac ( G ) → Jac ( G ) D By the above, we have that the restriction of ϕ to U is injective. Furthermore, since | im ( ϕ ) | | H | , we see that m | H | .We now wish to bound | V ( G ) | in terms of m . To do so, we construct a new graph G ′ from G , according to the following algorithm.(1) Choose any vertex of G of valency . Delete it, and draw an edge between its neighbors.(2) Repeat until there are no 2-valent vertices remaining.Note that even if G is simple, G ′ need not be. It is clear, however, that G and G ′ havethe same number of vertices with valency greater than , and that δ ( G ) = δ ( G ′ ) .By Lemma 29, we must have that e ′ = | E ( G ′ ) | is at most m · µ (since otherwise therewould necessarily be a vertex of G with valency greater than δ ). Each 2-valent vertex of G is uniquely associated with some edge of G ′ . If there are more than ( e ′ · µ ) divalentvertices in G , then at least µ of them are associated with a single edge of G ′ . In this case, → F IGURE
8. The transformation G G ′ G would contain a path P of length greater than µ , where each vertex of P has valency .This contradicts Lemma 33, so we have | V ( G ) | − m < mµ . If we let n H = | H | ( + µ ) , then | V ( G ) | < n H . (cid:3) Applying Corollary 30 and Proposition 32, we see that for sufficiently large k , we musthave | V ( G ) | > n H . This in turn implies that for sufficiently large k , ( Z /2 Z ) k × H is not theJacobian of any biconnected simple graph. We will use this fact to show that this resultholds generally, for all simple graphs. Proof of Theorem 6.
We proceed by induction on | H | . When | H | = or , Theorem 5 givesthe bound k H = . For | H | > , there must exist (by Proposition 34) an integer k ′ such that,if k > k ′ and Jac ( G ) ∼ = ( Z /2 Z ) k × H , then G is not biconnected.By the inductive hypothesis, for any proper subgroup H ′ ⊂ H , there exists an integer k ( H ′ ) such that for all k > k ( H ′ ) , no simple graph G ′ has Jac ( G ′ ) ∼ = ( Z /2 Z ) k × H ′ . Now,since H is finite, there are finitely many pairs of nontrivial proper subgroups H , H ⊂ H such that H × H ∼ = H . Define k ′′ = max { k ( H ) + k ( H ) : H , H nontrivial , H × H ∼ = H } . Now let k H = max ( k ′ , k ′′ ) . We wish to show that for all k > k H , if Jac ( G ) ∼ = ( Z /2 Z ) k × H ,then G is not simple. Let G be a graph with this Jacobian, and let k > k H . Since k > k ′ , G is not biconnected, so it must be the wedge sum of two graphs G and G . There mustthen exist integers k , k with k + k = k and groups H , H with H × H ∼ = H such that Jac ( G ) ∼ = ( Z /2 Z ) k × H , Jac ( G ) ∼ = ( Z /2 Z ) k × H . Without loss of generality, we may assume that neither G nor G is a tree, so that Jac ( G ) and Jac ( G ) are both nontrivial. If either H or H are trivial, then G (resp. G )would have Jacobian isomorphic to ( Z /2 Z ) k for k > 0 , contradicting Theorem 5.Finally, since k + k = k > k ′′ > k ( H ) + k ( H ) , we must have that either k > k ( H ) or k > k ( H ) . It follows that either G or G is not simple, so G is not simple. (cid:3) Further queries.
Analysis of the proof of Theorem 6 suggests that, if H ∼ = Z /p r Z for some prime p , then k H = O ( | H | p ) . In practice, it seems that much better boundsshould hold. For instance, we were unable to find any simple graph G where Jac ( G ) ∼ =( Z /2 Z ) k × H for any k > | H | .In some cases, it is possible to directly verify that certain groups do not arise as the Jaco-bian of any simple graph. Recall that a graph is 2-edge-connected if it remains connected fter the deletion of any edge. For a given m , while there are infinitely many isomorphismclasses of simple graphs with fewer than m spanning trees, at most finitely many of theseclasses represent 2-edge-connected graphs. This results from the fact that, for any vertex v on a 2-edge-connected graph, any divisor of the form v − v is v -reduced, and hencethere are at least as many spanning trees on the graph as there are vertices.By contracting bridges, any graph G may be uniquely associated to a 2-edge-connectedgraph with isomorphic Jacobian. For a given group H , therefore, it is possible to computethe Jacobian of all 2-edge-connected simple graphs with at most | H | spanning trees, andverify that H does or does not occur.Computer searches of this nature have led to the following: Proposition 35.
The following groups are not isomorphic to the Jacobian of any simple graph: • Z /2 Z × Z /4 Z , • ( Z /2 Z ) × Z /4 Z , • Z /2 Z × ( Z /4 Z ) . The key fact in the proof of the nonoccurence of groups with many factors of Z /2 Z seems to be the requirement that G is biconnected, rather than that G is simple. It has beenshown that, asymptotically, the probability that the Jacobian of a random graph is cyclicis relatively high [5]. We expect that the Jacobians of most graphs have a small number ofinvariant factors. Since random graphs are highly connected, we conjecture the following. Conjecture 36.
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Louis Gaudet D EPARTMENT OF M ATHEMATICS , R
UTGERS U NIVERSITY
E-mail address : [email protected] David Jensen D EPARTMENT OF M ATHEMATICS , U
NIVERSITY OF K ENTUCKY
E-mail address : [email protected] Dhruv Ranganathan D EPARTMENT OF M ATHEMATICS , M
ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY
E-mail address : [email protected] Nicholas Wawrykow D EPARTMENT OF M ATHEMATICS , U
NIVERSITY OF M ICHIGAN
E-mail address : [email protected] Theodore Weisman D EPARTMENT OF M ATHEMATICS , U
NIVERSITY OF T EXAS
E-mail address : [email protected]@math.utexas.edu