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

• n 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 = λ1 ≤ λ1 ≤ . . . ≤ λn. 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 χG(λ) = det(λIn − L(G)) the characteristic polynomial of a Laplacian of G. It can be presented as χG(λ) = c1λ + . . . + cn−1λn−1 + λn. The classical theorem by Kel’mans-Chelnokov (1975) states that the coefficient |ck| is equal to the number fk(n) of rooted spanning k-forests in graph G. We note that the values ck form alternating series. So, the total number of rooted spanning forests in graph G is FG(n) = f1 + f2 + . . . + fn = |c1 − c2 + c3 − . . . + (−1)n−1| = (−1)nχG(−1) = det(In + 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 Cn(s1, s2, . . . , sk) is a graph on n vertices v1, v2, v3, . . . , vn. The set of edges of G consists of elements vivi+sj, i = 1, . . . , n, j = 1, . . . , k. All indexes of vertices are taken by modulo n. From here on we will suppose that 1 ≤ s1 < s2 < . . . < sk < n

2.

For simplicity, we will use G instead of Cn(s1, s2, . . . , sk). The main aims of report are: 1◦ — to establish analytical formulas for the number of rooted spanning forests FG(n), 2◦ — to investigate asymptotic behaviour of such formulas with respect to n, 3◦ — to define some arithmetical properties of FG(n).

• I. Mednykh, L. Grunwald (NSU)

Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 4 / 14

The characteristic polynomial χG(λ)

We prove the theorem which express the characteristic polynomial χG(λ) as a product of some fixed (independent of n) number of multiples each of them are expressed through Tchebychev polynomials Tn(z) = cos(n arccos(z)) of degree n.

Theorem

The characteristic polynomial of a circulant graph G = Cn(s1, s2, . . . , sk) can be expressed in the form χG(λ) = (−1)n sk

sk

• j=1

(2Tn(wj(λ)) − 2), where wj(λ), j = 1, 2, . . . , sk are all the roots of the algebraic equation

k

• j=1

(2Tsj(w) − 2) = −λ.

• I. Mednykh, L. Grunwald (NSU)

Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 5 / 14

The characteristic polynomial χG(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), where L(z) = 2k −

k

• j=1

(zsj + z−sj) is a Laurent polynomial and T is a circulant n × n matrix circ(0, 1, 0, . . . , 0). Consider the characteristic polynomial of a graph G defined as χ(G, λ) = det(λ In − 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 T = diag(1, εn, . . . , εn−1

n

), where εn = e

2πi n . So, the matrix λ In − L(T) is

conjugated to a diagonal matrix λ In − L(T).

• I. Mednykh, L. Grunwald (NSU)

Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 6 / 14

The characteristic polynomial χG(λ)

The eigenvalues of the last matrix are easy to find. They are λj = λ − L(εj

n), j = 0, 1, . . . , n − 1.

If we set Pλ(z) = λ − L(z) we get χG(λ) =

n−1

• j=0

Pλ(εj

n).

We introduce polynomial Pλ(z) = zskPλ(z). It is a monic polynomial of a variable z of even degree 2sk. Polynomials Pλ(z) and Pλ(z) have the same set of roots. We note that Pλ(z) = Pλ( 1

z ). So we can write all the roots of

a polynomial Pλ(z) as z1, 1/z1, . . . , zsk, 1/zsk. At the same time the values wj = 1

2(zj + z−1 j

), j = 1, . . . , sk are all the roots of a polynomial Qλ(w) = λ +

k

• j=1

(2Tsj(w) − 2). To obtain this result we use the equality 1

2(zm + z−m) = Tm( 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 χG(λ)

All the values εj

n = e

2πij n , j = 0, 1, . . . , n − 1 are all the roots of the

polynomial zn − 1. We use the basic properties of resultant to get

n−1

• j=0
• Pλ(εj

n)

= Res( Pλ(z), zn − 1) = Res(zn − 1, Pλ(z)) =

• z :

Pλ(z)=0

(zn − 1) =

sk

• j=1

(zn

j − 1)(z−n j

− 1) =

sk

• j=1

(2 − zn

j − z−n j

) =

sk

• j=1

(2 − 2Tn(wj)).

• I. Mednykh, L. Grunwald (NSU)

Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 8 / 14

The characteristic polynomial χG(λ)

Taking into account

n−1

• j=0
• Pλ(εj

n) = n−1

• j=0

εj sk

n Pλ(εj n) = (−1)sk(n−1) n−1

• j=0

Pλ(εj

n),

we have the following χ(G, λ) =

n−1

• j=0

Pλ(εj

n) = (−1)sk(n−1) sk

• j=1

(2 − 2Tn(wj)) = (−1)n sk

sk

• j=1

(2Tn(wj) − 2). 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 FG(n).

As a direct consequence of the previous theorem we get

Theorem

The number of rooted spanning forests FG(n) in a circulant graph G = Cn(s1, s2, . . . , sk), 1 ≤ s1 < s2 < . . . < sk < n

2 is given by the formula

FG(n) =

sk

• p=1

|2Tn(wp) − 2|, where wp, p = 1, 2, . . . , sk are all the roots of the polynomial

k

• j=1

(2Tsj(w) − 2) = 1, and Ts(w) is a Tchebychev polynomial of the first kind. Proof: We set λ = −1 in the previous theorem and use the equality FG(n) = |χG(−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 FG(n).

Theorem

The number of rooted spanning forests in a circulant graph Cn(s1, s2, . . . , sk), 1 ≤ s1 < s2 < . . . < sk < n

2 has the following asymptotic

fG(n) ∼ An, as n → ∞, where A = exp( 1

0 log(1 + L(e2πit))dt) is a Mahler measure of a

polynomial 1 + L(z) and L(z) = 2k −

k

• i=1

(zsi + z−si).

• I. Mednykh, L. Grunwald (NSU)

Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 11 / 14

Proof for asymptotic of FG(n).

We have the equallity FG(n) =

sk

• p=1

|2Tn(wp) − 2|, where wp = 1

2(zp + z−1 p )

and zp, 1/zp, p = 1, . . . , sk are all the roots of the equation 1 + L(z) = 0. We note ϕ ∈ R, 1 + L(eiϕ) = 1 +

k

• j=1

(2 − 2 cos(sjϕ)) ≥ 1. Hence |zp| = 1 for all p. Replacing, if necessary, zp by 1/zp, we can assume that |zp| > 1, p = 1, . . . , sk. As 2Tn(wp) − 2 = zn

p + z−1 p

− 2 ∼ zn

p when n → ∞, we have sk

• p=1

|2 Tn(wp) − 2| ∼

sk

• p=1

|zp|n =

• 1+L(z)=0, |z|>1

|z|n = An, where A =

• 1+L(z)=0, |z|>1

|z| is a Mahler measure of a polynomial 1 + L(z). By Mahler measure’s properties A = exp( 1

0 log(1 + L(e2πit))dt).

• I. Mednykh, L. Grunwald (NSU)

Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 12 / 14

Arithmetics for the FG(n).

Theorem

Let FG(n) be the number of rooted spanning forests in a circulant graph Cn(s1, s2, . . . , sk), 1 ≤ s1 < s2 < . . . < sk < n 2. Set p to be equal to the number of odd elements in the sequence s1, s2, . . . , sk, and set q to be equal to the squarefree part of 4p + 1. Then there exist integer sequence a(n) such that 1◦ FG(n) = a(n)2, if n is odd, 2◦ FG(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

Examples

1◦ Cn = Cn(1) — cycle graph. We solve the equation 1 + 2 − 2T1(w) = 0. We get w = 3/2. Hence, FCn(n) = 2Tn(3/2) − 2. Also, FG(n) ∼

n→∞ ( 3+ √ 5 2

)n. 2◦ Cn(1, 2). We need to solve the equation 1 + 4 − 2T1(w) − 2T2(2) = 0. Its roots are w1 = 1

4(−1 +

√ 29) and w2 = 1

4(−1 −

√ 29). So FCn(1,2)(n) = |2Tn(w1) − 2| · |2Tn(w2) − 2| ∼

n→∞ An, where

A = 1

4(7 +

√ 5 +

• 38 + 14

√ 5) ≃ 4.3902568 . . . . 3◦ Cn(1, 3). The number of odd jumps p is equal to 2. From her we conclude 4p + 1 = 9 and q = 1. So, FCn(1,3) is always a full square.

• I. Mednykh, L. Grunwald (NSU)

Enumeration of rooted spanning forests in circulant graph Yichang Aug 2019 14 / 14