complexity and jacobians for cyclic coverings of a...

Complexity and Jacobians for cyclic coverings of a graph

Alexander MednykhSobolev Institute of Mathematics

Novosibirsk State University

Summer School for Inetnational conference and PhD-Master onGroups and Graphs, Desighs and Dynamics, Yichang, China

August 20, 2019

Alexander Mednykh (IM SB RAS) Complexity and Jacobians for cyclic coverings of a graph 20.08.2019 1 / 29

This is a part of joint research with Young Soo Kwon, Tomo Pisanski, IlyaMednykh and Madina Deryagina.

The notion of the Jacobian group of graph (also known as the sandpilegroup, critical group, Picard group, dollar group) was independently givenby many authors ( D. Dhar, R. Cori and D. Rossin, M. Baker and S.Norine, N. L. Biggs, R. Bacher, P. de la Harpe and T. Nagnibeda, N.L.Biggs, M. Kotani, T. Sunada). This is a very important algebraic invariantof a finite graph.In particular, the order of the Jacobian group coincides with the number ofspanning trees of a graph. The latter number is known for many largefamilies of graphs. But the structure of Jacobian for such families are stillunknown. The aim of the present presentation provide structure theoremsfor Jacobians of circulant graphs and some their generalisations.The Jacobian for graphs can be considered as a natural discrete analogueof Jacobian for Riemann surfaces.Also there is a close connection between the Jacobian and Laplacianoperator of a graph.


Jacobians of circulant graphs

We define Jacobian Jac(G ) of a graph G as the Abelian group generatedby flows satisfying the first and the second Kirchhoff laws. We illustratethis notion on the following simple example.



Complete graph K4

The first Kirchhoff law is given by the equations

L1 :

a+ b + c = 0;x − y − b = 0;y − z − c = 0;z − x − a = 0.



Complete graph K4

The second Kirchhoff law is given by the equations

L2 :

x + b − a = 0;y + c − b = 0;z + a− c = 0.



Now Jac(K4) = 〈a, b, c , x , y , z : L1, L2〉.Since by L2 : x = a− b, y = b − c, z = c − a we obtain

〈a, b, c : a+b+c = 0, a+b+c−4b = 0, a+b+c−4c = 0, a+b+c−4a = 0〉 =

〈a, b, c : a+ b + c = 0, 4a = 0, 4b = 0, 4c = 0〉 =

〈a, b : 4a = 0, 4b = 0〉 ∼= Z4 ⊕ Z4.

So we have Jac(K4) ∼= Z4 ⊕ Z4.



The graphs under consideration are supposed to be unoriented and finite.They may have loops, multiple edges and to be disconnected.Let auv be the number of edges between two given vertices u and v of G .The matrix A = A(G ) = [auv ]u,v∈V (G), is called the adjacency matrix ofthe graph G .Let d(v) denote the degree of v ∈ V (G ), d(v) =

∑u auv , and let

D = D(G ) be the diagonal matrix indexed by V (G ) and with dvv = d(v).The matrix L = L(G ) = D(G )− A(G ) is called the Laplacian matrix of G .It should be noted that loops have no influence on L(G ). The matrix L(G )is sometimes called the Kirchhoff matrix of G .It should be mentioned here that the rows and columns of graph matricesare indexed by the vertices of the graph, their order being unimportant.



Consider the Laplacian matrix L(G ) as a homomorphism ZV → ZV , whereV = |V (G )| is the number of vertices of G . Thencoker(L(G )) = ZV /im(L(G)) is an abelian group. Let

coker(L(G )) ∼= Zt1 ⊕ Zt2 ⊕ · · · ⊕ ZtV ,

be its Smith normal form satisfying ti∣∣ti+1, (1 ≤ i ≤ V ). If graph G is

connected then the groups Zt1 ,Zt1 , . . .ZtV−1 are finite and ZtV = Z. Inthis case,

Jac(G ) = Zt1 ⊕ Zt2 ⊕ · · · ⊕ ZtV−1is the Jacobian group of the graph G .Equivalently coker(L(G )) ∼= Jac(G )⊕ ZorJac(G ) is the torsion part of cokernel of L(G ).



Circulant graphs

Circulant graphs can be described in a few equivalent ways:

(a) The graph has an adjacency matrix that is a circulant matrix.(b) The automorphism group of the graph includes a cyclic subgroup that

acts transitively on the graph’s vertices.(c) The graph is a Cayley graph of a cyclic group.



Examples(a) The circulant graph Cn(s1, . . . , sk) with jumps s1, . . . , sk is defined as

the graph with n vertices labeled 0, 1, . . . , n − 1 where each vertex i isadjacent to 2k vertices i ± s1, . . . , i ± sk mod n.

(b) n-cycle graph Cn = Cn(1).(c) n-antiprism graph C2n(1, 2).(d) n-prism graph Yn = C2n(2, n), n odd.(e) The Moebius ladder graph Mn = C2n(1, n).(f) The complete graph Kn = Cn(1, 2, · · · , [n2 ]).(g) The complete bipartite graph Kn,n = Cn(1, 3, · · · , 2[n2 ] + 1).



The simplest possible circulant graphs with even degree of vertices arecyclic graphs Cn = Cn(1).Their Jacobians are cyclic groups Zn. The nextrepresentative of circulant graphs is the graph Cn(1, 2).

Circulant graph Cn(1, 2) for n = 6.



The structure of Jacobian of the graphs is given by the following theorem.

Theorem (Structure of Jac(Cn(1, 2)))Let A be the following matrix

A =

0 1 0 00 0 1 00 0 0 1−1 −1 4 −1

Then Jacobian of the circulant graph Cn(1, 2) is isomorphic to the torsionpart of cokernel of the operator

An − I4 : Z4 → Z4.



The following corollary is a consequence of the previous theorem.

CorollaryJacobian of the graph Cn(1, 2) is isomorphic to Z(n,Fn) ⊕ ZFn ⊕ Z[n,Fn],where (a, b) = GCD(a, b), [a, b] = LCM(a, b) and Fn - Fibonacci numbersdefined by recursion F1 = 1, F2 = 1, Fn+2 = Fn+1 + Fn, n ≥ 1.

Similar results can be obtained also for graphs Cn(1, 3) and Cn(2, 3). Inthese cases the structure of the Jacobians is expressed in terms of of realand imaginary parts of the Chebyshev polynomials Tn(1+i2 ), Un−1(

1+i2 ) and

Tn(3+i√3

4 ), Un−1(3+i√3

4 ) respectively. Recall that

Tn(x) = cos(n arccos x) and Un−1(x) =sin(n arccos x)

sin(arccos x).



Consider the family of circulant graphs Cn(1, 3).

Case n = 7 is show below.

��



Theorem

Jacobian Jac(Cn(1, 3)) is isomorphic to Zd1 ⊕ Zd2 ⊕ · · · ⊕ Zd5 , wheredi∣∣di+1, (1 ≤ i ≤ 5). Here d1 = (n, d), d2 = d , if 4 is not divisor of n;

otherwise d1 = (n, d)/2, d2 = d/2, if n/4 is even andd1 = (n, d)/4, d2 = d/4, if n/4 is odd. Set d = GCD(s, t, u, v) ands, t, u, v are integers defined by the equations s + i t = 2Tn(1+i2 )− 2 andu + i v = Un−1(

1+i2 ). Moreover, the order of the group Jac(Cn(1, 3)) is

equal to n(s2 + t2)/10.

RemarkIn the above theorem the numbers di , (3 ≤ i ≤ 5) can be expressedthrough n, s, t, u, v . But the respective formulas are rather large andcomplicated.


n Jacobian Jac(Cn(1, 3))7 Z13 ⊕ Z918 Z4 ⊕ Z4 ⊕ Z4 ⊕ Z4 ⊕ Z169 Z37 ⊕ Z33310 Z3 ⊕ Z15 ⊕ Z15 ⊕ Z6011 Z109 ⊕ Z119912 Z2 ⊕ Z130 ⊕ Z156013 Z313 ⊕ Z406914 Z337 ⊕ Z1055615 Z5 ⊕ Z905 ⊕ Z271516 Z8 ⊕ Z8 ⊕ Z8 ⊕ Z136 ⊕ Z54417 Z21617 ⊕ Z4448918 Z3145 ⊕ Z11322019 Z7561 ⊕ Z14365920 Z3 ⊕ Z30 ⊕ Z3030 ⊕ Z1212021 Z41 ⊕ Z41 ⊕ Z533 ⊕ Z1119322 Z26269 ⊕ Z115583623 Z63157 ⊕ Z1452611



More general result is given by the following theorem.

TheoremLet G = Cn(s1, s2, s3, . . . , sk), where 1 ≤ s1 < s2 < . . . < sk < n2 be acirculant graph of even degree. Let A = As1,s2,s3,...,sk be companion matrix

of the Laurent polynomial L(z) = 2k −k∑

j=1(zsj + z−sj ). Then the Jac(G ) is

isomorphic to the torsion part of cokernel of An − I2sk : Z2sk → Z2sk .Moreover, the rank of Jac(G ) is at least 2 and at most 2sk − 1. The bothestimates are sharp.


Jacobians of circulant graphs with odd degree

One of the simplest examples of circulant graph with odd degree of verticesis the Moebius ladder graph C2n(1, n).

TheoremJacobian Jac(M(n)) of the Moebius band M(n) is isomorphic toZ(n,Hm) ⊕ZHm ⊕Z3{n,Hm}, if n = 2m+ 1 is odd, Z(n,Tm) ⊕ZTm ⊕Z2{n,Tm},n = 2m and m is even, and Z(n,Tm)/2 ⊕ Z2Tm ⊕ Z2{n,Tm}, if n = 2m and mis odd, where (l ,m) = GCD(l ,m), {l ,m} = LCM(l ,m), Hm = Tm +Um−1,and Tm = Tm(2), Um−1 = Um−1(2) are the Chebyshev polynomials of thefirst and the second type respectively.

This theorem can be considered as a refined version of the results obtainedearlier by P. Cheng, Y. Hou, C. Woo (2006) and I.A. Mednykh, M.A.Deryagina (2011).


Jacobians of circulant graphs with odd degree

The general result is given by the following theorem.

TheoremLet C2n(s1, s2, s3, . . . , sk , n), 1 ≤ s1 ≤ . . . ≤ sk < n be a circulant graph ofodd degree. Let A = As1,s2,s3,...,sk be companion matrix of the Laurent

polynomial Q(z) = L2(z)− 1, where L(z) = 2k +1−k∑

j=1(zsj + z−sj ). Then

the Jacobian group of circulant graph C2n(s1, s2, s3, . . . , sk , n) is isomorphicto the torsion part of the cokernel of operator L(A)−An : Z4sk → Z4sk .


Counting spanning trees for circulant graphs

Recall that the number of spanning trees τ(G ) of a graph G coinsides with|Jac(G )|. The above mentioned results leads to explicit formulae for τ(G )for any circulant graph G . We restrict ourself only on the followingproperties of τ(G ). Recall that any positive integer p can be uniquelyrepresented in the form p = q r2, where p and q are positive integers and qis square-free. We will call q the square-free part of p.

Theorem

Let τ(n) be the number of spanning trees of the circulant graphCn(s1, s2, s3, . . . , sk), 1 ≤ s1 < s2 < . . . < sk < n2 . Denote by p the numberof odd elements in the sequence s1, s2, s3, . . . , sk and let q be thesquare-free part of p. Then there exists an integer sequence a(n) such that10 τ(n) = n a(n)2, if n is odd;20 τ(n) = q n a(n)2, if n is even.


Counting spanning trees for circulant graphs

In a similar way, one can get

TheoremLet τ(n) be the number of spanning trees of the circulant graphC2n(s1, s2, s3, . . . , sk , n), 1 ≤ s1 < s2 < . . . < sk < n2 . Denote by p thenumber of odd elements in the sequence s1, s2, s3, . . . , sk . Let q be thesquare-free part of 2p and r be the square-free part of 2p + 1. Then thereexists an integer sequence a(n) such that10. τ(n) = r n a(n)2, if n is odd;20. τ(n) = q n a(n)2, if n is even.

For example, for the Moebius ladder C2n(1, n) there exists an integersequence a(n) such that τ(n) = 3n a(n)2 if n is odd, and τ(n) = 2n a(n)2

if n is even. More precisely, in the above notation a(2m + 1) = Hm anda(2m) = Tm.


Asimptotic for the number of spanning trees

Theorem

The number of spanning trees of the circulant graph Cn(s1, s2, s3, . . . , sk)has the following asymptotic

τ(n) ∼ nqAn, as n→∞,

where q = s21 + s22 + . . .+ s

2k and

A = exp(

∫ 10

log |L(e2πit)|dt)

is the Mahler measure of Laurent polynomial L(z) = 2k −k∑

i=1(zsi + z−si ).



The next theorem can be proved by similar arguments.

Theorem

The number of spanning trees of the circulant graphC2n(s1, s2, s3, . . . , sk , n), has the following asymptotic

τ(n) ∼ n2 q

Kn, as n→∞,

where q = s21 + s22 + . . .+ s

2k , and

K = exp(

∫ 10

log |Q(e2πit |dt)

is the Mahler measure of Laurent polynomial Q(z) = L2(z)− 1, where

L(z) = 2k + 1−k∑

j=1(zsj + z−sj ).



We illustrate the obtained results by some examples.

Graph Cn(1, 2, 3). Here A1,2,3 = 12(2 +√7 +

√7 + 4

√7) ≈ 4.42 and

τ(n) ∼ n14An1,2,3, n→∞. Also, there exists an integer sequence a(n) such

that τ(n) = n a(n)2 if n is odd, and τ(n) = 2n a(n)2 if n is even.

Graph C2n(1, 2, n)K1,2 =

14(3+

√5)(4+

√3+√15 + 8

√3) ≈ 14.54, τ(n) ∼ n10 K

n1,2, n→∞.

There exists an integer sequence a(n) such that τ(n) = 3n a(n)2 if n is oddand τ(n) = 2n a(n)2 if n is even.


Further generalisations

The results discussed in the lecture can be also obtained for generalisedPetersen graphs, I -,Y - and H-graphs. These are defined as cyclic branchedcovering over the graphs of shape I ,Y and H, respectively. We illustrateour results in the progress (2017+) by the following two theorems.

Theorem

Jacobian group Jn of the Y -graph Y (n; 1, 1, 1) for n ≥ 4 has the followingstructure10 Jn ' Zn−43 ⊕ Z3n ⊕ Z2Ln ⊕ Z

23Ln, if n is odd,

20 Jn ' Zn−43 ⊕ Z3n ⊕ ZFn ⊕ Z3Fn ⊕ Z5Fn ⊕ Z15Ln , n is even,where Ln and Fn are the Lucas and the Fibonacci numbers respectively.


Jacobians of Haar graphs

The following is a recent result joint with Ilya Mednykh and TomasPisanski (2019+).

Theorem

Let L be Laplacian of the graph of H(Zn, {0, 1, 2}). Then

coker L ∼= coker (An − I ),

where A is the following matrix

A =

0 0 1 00 0 0 18 −3 −1 −33 −1 0 −1

.


As a consequence, we have the following corollary.

Corollary

Jacobian of the Haar graph H(Zn, {0, 1, 2}) is isomorphic toZt1 ⊕ Zt2 ⊕ Zt3 , where(1) t1 = gcd(n, a(n)), t2 = a(n), t3 = lcm(n, 3a(n)), where

a(n) =√

2/3Tn(√

3/2) if n is odd(2) t1 = gcd(n/2, b(n)), t2 = gcd(n/2, 2)b(n), t3 = lcm(2n, 6b(n)), where

b(n) =√

1/6Un−1(√3/2) if n is even.

Here, Tn(x) and Un−1(x) are the Chebyshev polynomials of the first andsecond - kind respectively.


Theorem

Let L be Laplacian of the graph of H(Zn, {0, 1, 2, 3}). Then

coker L ∼= coker (An − I ),

where A is the following matrix

A =

0 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 115 −4 −1 −4 −1 −44 −1 0 −1 0 −1

.

The matrix An − I has an explicit form in terms of the Chebyshevpolynomials Tn(−1 + i) and Un−1(−1 + i). This gives us a possibility toprove the following result.


As a consequence, we have the following corollary.

Corollary

(i) The number of spanning trees in the graph H(Zn, {0, 1, 2, 3}) is givesby the formula

τ(n) =4n

5(Tn(−1 + i)− 1)(Tn(−1− i)− 1).

(ii) Jacobian Jac (H) of the graph H = H(Zn, {0, 1, 2, 3}) has the followingstructure

Jac (H) ∼= Zd1 ⊕ Zd2 ⊕ Zd3 ⊕ Zd4 ⊕ Zd5 ,

where d1|d2|d3|d4|d5 and d1d2d3d4d5 = τ(n).Moreover, if d = gcd(Re(Tn(−1 + i)− 1), Im(Tn(−1 + i)),Re(Un−1(−1 +i)), Im(Un−1(−1 + i))), then d1 = gcd(n, d)/2, d2 = d/2 if i ≡ 2(mod 4) and d1 = gcd(n, d), d2 = d otherwise.


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