Complexity and Jacobians for cyclic coverings of a graph Alexander Mednykh Sobolev Institute of Mathematics Novosibirsk State University Summer School for Inetnational conference and PhD-Master on Groups 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, Ilya Mednykh and Madina Deryagina. The notion of the Jacobian group of graph (also known as the sandpile group, critical group, Picard group, dollar group) was independently given by 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 invariant of a finite graph. In particular, the order of the Jacobian group coincides with the number of spanning trees of a graph. The latter number is known for many large families of graphs. But the structure of Jacobian for such families are still unknown. The aim of the present presentation provide structure theorems for Jacobians of circulant graphs and some their generalisations. The Jacobian for graphs can be considered as a natural discrete analogue of Jacobian for Riemann surfaces. Also there is a close connection between the Jacobian and Laplacian operator of a graph. Alexander Mednykh (IM SB RAS) Complexity and Jacobians for cyclic coverings of a graph 20.08.2019 2 / 29
Jacobians of circulant graphs We define Jacobian Jac ( G ) of a graph G as the Abelian group generated by flows satisfying the first and the second Kirchhoff laws. We illustrate this notion on the following simple example. Alexander Mednykh (IM SB RAS) Complexity and Jacobians for cyclic coverings of a graph 20.08.2019 3 / 29
Jacobians of circulant graphs Complete graph K 4 The first Kirchhoff law is given by the equations a + b + c = 0; x − y − b = 0; L 1 : y − z − c = 0; z − x − a = 0 . Alexander Mednykh (IM SB RAS) Complexity and Jacobians for cyclic coverings of a graph 20.08.2019 4 / 29
Jacobians of circulant graphs Complete graph K 4 The second Kirchhoff law is given by the equations x + b − a = 0; L 2 : y + c − b = 0; z + a − c = 0 . Alexander Mednykh (IM SB RAS) Complexity and Jacobians for cyclic coverings of a graph 20.08.2019 5 / 29
Jacobians of circulant graphs Now Jac ( K 4 ) = � a , b , c , x , y , z : L 1 , L 2 � . Since by L 2 : x = a − b , y = b − c , z = c − a we obtain � a , b , c : a + b + c = 0 , a + b + c − 4 b = 0 , a + b + c − 4 c = 0 , a + b + c − 4 a = 0 � = � a , b , c : a + b + c = 0 , 4 a = 0 , 4 b = 0 , 4 c = 0 � = � a , b : 4 a = 0 , 4 b = 0 � ∼ = Z 4 ⊕ Z 4 . So we have Jac ( K 4 ) ∼ = Z 4 ⊕ Z 4 . Alexander Mednykh (IM SB RAS) Complexity and Jacobians for cyclic coverings of a graph 20.08.2019 6 / 29
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 a uv be the number of edges between two given vertices u and v of G . The matrix A = A ( G ) = [ a uv ] u , v ∈ V ( G ) , is called the adjacency matrix of the graph G . Let d ( v ) denote the degree of v ∈ V ( G ) , d ( v ) = � u a uv , and let D = D ( G ) be the diagonal matrix indexed by V ( G ) and with d vv = 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 matrices are indexed by the vertices of the graph, their order being unimportant. Alexander Mednykh (IM SB RAS) Complexity and Jacobians for cyclic coverings of a graph 20.08.2019 7 / 29
Jacobians of circulant graphs Consider the Laplacian matrix L ( G ) as a homomorphism Z V → Z V , where V = | V ( G ) | is the number of vertices of G . Then coker ( L ( G )) = Z V / im ( L ( G )) is an abelian group. Let coker ( L ( G )) ∼ = Z t 1 ⊕ Z t 2 ⊕ · · · ⊕ Z t V , � be its Smith normal form satisfying t i � t i +1 , (1 ≤ i ≤ V ) . If graph G is connected then the groups Z t 1 , Z t 1 , . . . Z t V − 1 are finite and Z t V = Z . In this case, Jac ( G ) = Z t 1 ⊕ Z t 2 ⊕ · · · ⊕ Z t V − 1 is the Jacobian group of the graph G . Equivalently coker ( L ( G )) ∼ = Jac ( G ) ⊕ Z or Jac ( 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
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. Alexander Mednykh (IM SB RAS) Complexity and Jacobians for cyclic coverings of a graph 20.08.2019 9 / 29
Jacobians of circulant graphs Examples (a) The circulant graph C n ( s 1 , . . . , s k ) with jumps s 1 , . . . , s k is defined as the graph with n vertices labeled 0 , 1 , . . . , n − 1 where each vertex i is adjacent to 2 k vertices i ± s 1 , . . . , i ± s k mod n . (b) n -cycle graph C n = C n (1) . (c) n -antiprism graph C 2 n (1 , 2) . (d) n -prism graph Y n = C 2 n (2 , n ) , n odd. (e) The Moebius ladder graph M n = C 2 n (1 , n ) . (f) The complete graph K n = C n (1 , 2 , · · · , [ n 2 ]) . (g) The complete bipartite graph K n , n = C n (1 , 3 , · · · , 2[ n 2 ] + 1) . Alexander Mednykh (IM SB RAS) Complexity and Jacobians for cyclic coverings of a graph 20.08.2019 10 / 29
Jacobians of circulant graphs The simplest possible circulant graphs with even degree of vertices are cyclic graphs C n = C n (1) . Their Jacobians are cyclic groups Z n . The next representative of circulant graphs is the graph C n (1 , 2) . Circulant graph C n (1 , 2) for n = 6 . Alexander Mednykh (IM SB RAS) Complexity and Jacobians for cyclic coverings of a graph 20.08.2019 11 / 29
Jacobians of circulant graphs The structure of Jacobian of the graphs is given by the following theorem. Theorem (Structure of Jac ( C n (1 , 2)) ) Let A be the following matrix 0 1 0 0 0 0 1 0 A = 0 0 0 1 − 1 − 1 4 − 1 Then Jacobian of the circulant graph C n (1 , 2) is isomorphic to the torsion part of cokernel of the operator A n − I 4 : Z 4 → Z 4 . Alexander Mednykh (IM SB RAS) Complexity and Jacobians for cyclic coverings of a graph 20.08.2019 12 / 29
Jacobians of circulant graphs The following corollary is a consequence of the previous theorem. Corollary Jacobian of the graph C n (1 , 2) is isomorphic to Z ( n , F n ) ⊕ Z F n ⊕ Z [ n , F n ] , where ( a , b ) = GCD ( a , b ) , [ a , b ] = LCM ( a , b ) and F n - Fibonacci numbers defined by recursion F 1 = 1 , F 2 = 1 , F n +2 = F n +1 + F n , n ≥ 1 . Similar results can be obtained also for graphs C n (1 , 3) and C n (2 , 3) . In these cases the structure of the Jacobians is expressed in terms of of real and imaginary parts of the Chebyshev polynomials T n ( 1+ i 2 ) , U n − 1 ( 1+ i 2 ) and √ √ T n ( 3+ i 3 ) , U n − 1 ( 3+ i 3 ) respectively. Recall that 4 4 T n ( x ) = cos( n arccos x ) and U n − 1 ( x ) = sin( n arccos x ) sin( arccos x ) . Alexander Mednykh (IM SB RAS) Complexity and Jacobians for cyclic coverings of a graph 20.08.2019 13 / 29
Jacobians of circulant graphs Consider the family of circulant graphs C n (1 , 3) . Case n = 7 is show below. ������� Alexander Mednykh (IM SB RAS) Complexity and Jacobians for cyclic coverings of a graph 20.08.2019 14 / 29
Jacobians of circulant graphs Theorem Jacobian Jac ( C n (1 , 3)) is isomorphic to Z d 1 ⊕ Z d 2 ⊕ · · · ⊕ Z d 5 , where � d i � d i +1 , (1 ≤ i ≤ 5) . Here d 1 = ( n , d ) , d 2 = d , if 4 is not divisor of n ; otherwise d 1 = ( n , d ) / 2 , d 2 = d / 2 , if n / 4 is even and d 1 = ( n , d ) / 4 , d 2 = d / 4 , if n / 4 is odd. Set d = GCD ( s , t , u , v ) and s , t , u , v are integers defined by the equations s + i t = 2 T n ( 1+ i 2 ) − 2 and u + i v = U n − 1 ( 1+ i 2 ) . Moreover, the order of the group Jac ( C n (1 , 3)) is equal to n ( s 2 + t 2 ) / 10 . Remark In the above theorem the numbers d i , (3 ≤ i ≤ 5) can be expressed through n , s , t , u , v . But the respective formulas are rather large and complicated. Alexander Mednykh (IM SB RAS) Complexity and Jacobians for cyclic coverings of a graph 20.08.2019 15 / 29
n Jacobian Jac ( C n (1 , 3)) 7 Z 13 ⊕ Z 91 8 Z 4 ⊕ Z 4 ⊕ Z 4 ⊕ Z 4 ⊕ Z 16 9 Z 37 ⊕ Z 333 10 Z 3 ⊕ Z 15 ⊕ Z 15 ⊕ Z 60 11 Z 109 ⊕ Z 1199 12 Z 2 ⊕ Z 130 ⊕ Z 1560 13 Z 313 ⊕ Z 4069 14 Z 337 ⊕ Z 10556 15 Z 5 ⊕ Z 905 ⊕ Z 2715 16 Z 8 ⊕ Z 8 ⊕ Z 8 ⊕ Z 136 ⊕ Z 544 17 Z 21617 ⊕ Z 44489 18 Z 3145 ⊕ Z 113220 19 Z 7561 ⊕ Z 143659 20 Z 3 ⊕ Z 30 ⊕ Z 3030 ⊕ Z 12120 21 Z 41 ⊕ Z 41 ⊕ Z 533 ⊕ Z 11193 22 Z 26269 ⊕ Z 1155836 23 Z 63157 ⊕ Z 1452611 Alexander Mednykh (IM SB RAS) Complexity and Jacobians for cyclic coverings of a graph 20.08.2019 16 / 29
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