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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
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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.
Alexander Mednykh (IM SB RAS) Complexity and Jacobians for
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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.
Alexander Mednykh (IM SB RAS) Complexity and Jacobians for
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Jacobians of circulant graphs
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.
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Jacobians of circulant graphs
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.
Alexander Mednykh (IM SB RAS) Complexity and Jacobians for
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Jacobians of circulant graphs
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.
Alexander Mednykh (IM SB RAS) Complexity and Jacobians for
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Jacobians of circulant graphs
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.
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Jacobians of circulant graphs
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 ).
Alexander Mednykh (IM SB RAS) Complexity and Jacobians for
cyclic coverings of a graph 20.08.2019 8 / 29
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Jacobians of circulant graphs
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.
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Jacobians of circulant graphs
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).
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Jacobians of circulant graphs
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.
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Jacobians of circulant graphs
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.
Alexander Mednykh (IM SB RAS) Complexity and Jacobians for
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Jacobians of circulant graphs
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).
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Jacobians of circulant graphs
Consider the family of circulant graphs Cn(1, 3).
Case n = 7 is show below.
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Jacobians of circulant graphs
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.
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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
Alexander Mednykh (IM SB RAS) Complexity and Jacobians for
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Jacobians of circulant graphs
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.
Alexander Mednykh (IM SB RAS) Complexity and Jacobians for
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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).
Alexander Mednykh (IM SB RAS) Complexity and Jacobians for
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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 .
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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.
Alexander Mednykh (IM SB RAS) Complexity and Jacobians for
cyclic coverings of a graph 20.08.2019 20 / 29
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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.
Alexander Mednykh (IM SB RAS) Complexity and Jacobians for
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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 ).
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Asimptotic for the number of spanning trees
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 ).
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Asimptotic for the number of spanning trees
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.
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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.
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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
.
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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.
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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.
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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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