Enumeration of rooted spanning forests in circulant graph I. - PowerPoint PPT Presentation

Enumeration of rooted spanning forests in circulant graph I. Mednykh, L. Grunwald Sobolev Institute of Mathematics, Novosibirsk, Russia Novosibirsk State University, Russia The international conference and PhD-master summer school on Groups and Graphs, Designs and Dynamics Yichang, 12 - 25 Aug, 2019 22 August 2019 I. Mednykh, L. Grunwald (NSU) Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 1 / 14

Preliminary facts Let G be a finite undirected graph on n vertices. Subgraph of G without cycles is called forest . Connected component of a forest is called tree . The forest is called spanning if its set of vertices coincides with all vertices of G . The forest is called rooted if for each tree we choose a root , that is a marked vertex. The forest with k trees we call k -forest . The Laplacian of a graph G is a matrix L ( G ) = D ( G ) − A ( G ) . Here D ( G ) is a diagonal matrix with degrees of vertices of G on a diagonal. A ( G ) is adjacency matrix of a graph G . Recall that the matrix L ( G ) is always degenerated and nonnegative definite. That is L ( G ) ≥ 0 . Its spectrum is 0 = λ 1 ≤ λ 1 ≤ . . . ≤ λ n . The number of zero-valued eigenvalues of L ( G ) coincides with the number of connected components of G . I. Mednykh, L. Grunwald (NSU) Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 2 / 14

Preliminary facts Denote by χ G ( λ ) = det( λ I n − L ( G )) the characteristic polynomial of a Laplacian of G . It can be presented as χ G ( λ ) = c 1 λ + . . . + c n − 1 λ n − 1 + λ n . The classical theorem by Kel’mans-Chelnokov (1975) states that the coefficient | c k | is equal to the number f k ( n ) of rooted spanning k -forests in graph G . We note that the values c k form alternating series. So, the total number of rooted spanning forests in graph G is f 1 + f 2 + . . . + f n = | c 1 − c 2 + c 3 − . . . + ( − 1) n − 1 | F G ( n ) = ( − 1) n χ G ( − 1) = det( I n + L ( G )) . = This result was obtained independently by many authors (P. Chebatorev, E. Shamis, O. Knill and others). I. Mednykh, L. Grunwald (NSU) Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 3 / 14

Preliminary facts The circulant graph C n ( s 1 , s 2 , . . . , s k ) is a graph on n vertices v 1 , v 2 , v 3 , . . . , v n . The set of edges of G consists of elements v i v i + s j , i = 1 , . . . , n , j = 1 , . . . , k . All indexes of vertices are taken by modulo n . From here on we will suppose that 1 ≤ s 1 < s 2 < . . . < s k < n 2 . For simplicity, we will use G instead of C n ( s 1 , s 2 , . . . , s k ) . The main aims of report are: 1 ◦ — to establish analytical formulas for the number of rooted spanning forests F G ( n ) , 2 ◦ — to investigate asymptotic behaviour of such formulas with respect to n , 3 ◦ — to define some arithmetical properties of F G ( n ) . I. Mednykh, L. Grunwald (NSU) Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 4 / 14

The characteristic polynomial χ G ( λ ) We prove the theorem which express the characteristic polynomial χ G ( λ ) as a product of some fixed (independent of n ) number of multiples each of them are expressed through Tchebychev polynomials T n ( z ) = cos( n arccos( z )) of degree n . Theorem The characteristic polynomial of a circulant graph G = C n ( s 1 , s 2 , . . . , s k ) can be expressed in the form s k � χ G ( λ ) = ( − 1) n s k (2 T n ( w j ( λ )) − 2) , j =1 where w j ( λ ) , j = 1 , 2 , . . . , s k are all the roots of the algebraic equation k � (2 T s j ( w ) − 2) = − λ. j =1 I. Mednykh, L. Grunwald (NSU) Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 5 / 14

The characteristic polynomial χ G ( λ ) The idea of the proof is based on the following argumentations. The Laplacian of a a circulant graph G can be presented in the form L ( T ) , � k ( z s j + z − s j ) is a Laurent polynomial and T is a where L ( z ) = 2 k − j =1 circulant n × n matrix circ (0 , 1 , 0 , . . . , 0) . Consider the characteristic polynomial of a graph G defined as χ ( G , λ ) = det( λ I n − L ( T )) . Our aim is to express this determinant as a product of a fixed number of multiples each of them is a certain Tchebychev polynomial of degree n . Note that matrix T is conjugate to a diagonal matrix 2 π i T = diag (1 , ε n , . . . , ε n − 1 n . So, the matrix λ I n − L ( T ) is ) , where ε n = e n conjugated to a diagonal matrix λ I n − L ( T ) . I. Mednykh, L. Grunwald (NSU) Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 6 / 14

The characteristic polynomial χ G ( λ ) The eigenvalues of the last matrix are easy to find. They are λ j = λ − L ( ε j n ) , j = 0 , 1 , . . . , n − 1 . n − 1 � P λ ( ε j If we set P λ ( z ) = λ − L ( z ) we get χ G ( λ ) = n ) . j =0 We introduce polynomial � P λ ( z ) = z s k P λ ( z ) . It is a monic polynomial of a variable z of even degree 2 s k . Polynomials � P λ ( z ) and P λ ( z ) have the same set of roots. We note that P λ ( z ) = P λ ( 1 z ) . So we can write all the roots of a polynomial � P λ ( z ) as z 1 , 1 / z 1 , . . . , z s k , 1 / z s k . At the same time the values w j = 1 2 ( z j + z − 1 ) , j = 1 , . . . , s k are all the j � k roots of a polynomial Q λ ( w ) = λ + (2 T s j ( w ) − 2) . To obtain this result j =1 2 ( z m + z − m ) = T m ( 1 we use the equality 1 2 ( z + z − 1 )) . I. Mednykh, L. Grunwald (NSU) Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 7 / 14

The characteristic polynomial χ G ( λ ) 2 π i j All the values ε j n , j = 0 , 1 , . . . , n − 1 are all the roots of the n = e polynomial z n − 1 . We use the basic properties of resultant to get n − 1 � P λ ( z ) , z n − 1) = Res ( z n − 1 , � P λ ( ε j � Res ( � n ) = P λ ( z )) j =0 s k � � ( z n − 1) = j − 1)( z − n ( z n = − 1) j j =1 z : � P λ ( z )=0 s k s k � � (2 − z n j − z − n = ) = (2 − 2 T n ( w j )) . j j =1 j =1 I. Mednykh, L. Grunwald (NSU) Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 8 / 14

The characteristic polynomial χ G ( λ ) n − 1 n − 1 n − 1 � � � P λ ( ε j � ε j s k n P λ ( ε j P λ ( ε j n ) = ( − 1) s k ( n − 1) Taking into account n ) = n ) , j =0 j =0 j =0 we have the following n − 1 s k � � n ) = ( − 1) s k ( n − 1) P λ ( ε j χ ( G , λ ) = (2 − 2 T n ( w j )) j =0 j =1 s k � ( − 1) n s k = (2 T n ( w j ) − 2) . j =1 So theorem is proved. I. Mednykh, L. Grunwald (NSU) Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 9 / 14

The formula for the number of rooted spanning forests F G ( n ) . As a direct consequence of the previous theorem we get Theorem The number of rooted spanning forests F G ( n ) in a circulant graph G = C n ( s 1 , s 2 , . . . , s k ) , 1 ≤ s 1 < s 2 < . . . < s k < n 2 is given by the formula s k � F G ( n ) = | 2 T n ( w p ) − 2 | , p =1 where w p , p = 1 , 2 , . . . , s k are all the roots of the polynomial � k (2 T s j ( w ) − 2) = 1 , and T s ( w ) is a Tchebychev polynomial of the first j =1 kind. Proof: We set λ = − 1 in the previous theorem and use the equality F G ( n ) = | χ G ( − 1) | . I. Mednykh, L. Grunwald (NSU) Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 10 / 14

Asymptotic for the number of rooted spanning forests F G ( n ) . Theorem The number of rooted spanning forests in a circulant graph C n ( s 1 , s 2 , . . . , s k ) , 1 ≤ s 1 < s 2 < . . . < s k < n 2 has the following asymptotic f G ( n ) ∼ A n , as n → ∞ , � 1 0 log(1 + L ( e 2 π i t )) dt ) is a Mahler measure of a where A = exp( � k ( z s i + z − s i ) . polynomial 1 + L ( z ) and L ( z ) = 2 k − i =1 I. Mednykh, L. Grunwald (NSU) Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 11 / 14

Proof for asymptotic of F G ( n ) . � s k | 2 T n ( w p ) − 2 | , where w p = 1 2 ( z p + z − 1 We have the equallity F G ( n ) = p ) p =1 and z p , 1 / z p , p = 1 , . . . , s k are all the roots of the equation 1 + L ( z ) = 0 . � k We note ϕ ∈ R , 1 + L ( e i ϕ ) = 1 + (2 − 2 cos( s j ϕ )) ≥ 1 . Hence | z p | � = 1 j =1 for all p . Replacing, if necessary, z p by 1 / z p , we can assume that | z p | > 1 , p = 1 , . . . , s k . As 2 T n ( w p ) − 2 = z n p + z − 1 − 2 ∼ z n p when n → ∞ , we have p s k s k � � � | z p | n = | z | n = A n , | 2 T n ( w p ) − 2 | ∼ p =1 p =1 1+ L ( z )=0 , | z | > 1 � where A = | z | is a Mahler measure of a polynomial 1 + L ( z ) . 1+ L ( z )=0 , | z | > 1 � 1 0 log(1 + L ( e 2 π i t )) dt ) . By Mahler measure’s properties A = exp( I. Mednykh, L. Grunwald (NSU) Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 12 / 14

Arithmetics for the F G ( n ) . Theorem Let F G ( n ) be the number of rooted spanning forests in a circulant graph C n ( s 1 , s 2 , . . . , s k ) , 1 ≤ s 1 < s 2 < . . . < s k < n 2 . Set p to be equal to the number of odd elements in the sequence s 1 , s 2 , . . . , s k , and set q to be equal to the squarefree part of 4 p + 1 . Then there exist integer sequence a ( n ) such that 1 ◦ F G ( n ) = a ( n ) 2 , if n is odd, 2 ◦ F G ( n ) = q a ( n ) 2 , if n is even. I. Mednykh, L. Grunwald (NSU) Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 13 / 14