ON THE JACOBIAN GROUP OF A CONE OVER A CIRCULANT GRAPH

Grunwald L. A.; Mednykh I. A.

Математические заметки СВФУ Апрель—июнь, 2021. Том 28, № 2

UDC 517.545+517.962.2+519.173

ON THE JACOBIAN GROUP OF A CONE

OVER A CIRCULANT GRAPH L. A. Grunwald and I. A. Mednykh

Abstract. For any given graph G, consider the graph G which is a cone over G. We study two important invariants of such a cone, namely, the complexity (the number of spanning trees) and the Jacobian of the graph. We prove that complexity of graph G coincides with the number of rooted spanning forests in G and the Jacobian of G is isomorphic to the cokernel of the operator I + L(G), where L(G) is the Laplacian of G and I is the identity matrix. As a consequence, one can calculate the complexity of G as det(I + L(G)).

As an application, we establish general structural theorems for the Jacobian of G in the case when G is a circulant graph or cobordism of two circulant graphs.

DOI: 10.25587/SVFU.2021.32.84.006 Keywords: spanning tree, spanning forest, circulant graph, Laplacian matrix, cone over graph, Chebyshev polynomial.

1. Introduction

A spanning tree in a finite connected graph G is defined as a subgraph of G that contains all vertices of G and has no cycles. The number of spanning trees of the graph G is also called a complexity of G. It is a very important graph invariant and along with the pioneers in this field [1], formulas were found for some special graphs such as the wheel [2], fan [3], ladder [4], Mobius ladder [5], lattice [6], prism [7] and anti-prism [8]. However, one of the most significant and general results is the Kirchhoff matrix-tree theorem [9] which states that the complexity of G can be expressed as the product of nonzero Laplacian eigenvalues of G, divided by the number of its vertices. In this paper, we will also apply the idea [2] of using Chebyshev polynomials for counting various invariants of graphs arose.

Also, no less interesting invariant of a graph is the number of rooted spanning forests in a graph G. According to the classical result [10], this value can be found as determinant det(1+L(G)). Here, L(G) is the Laplacian matrix of graph G. However, not many explicit formulas are known. One of the first results was obtained by O. Knill [11], who found the analytical formula of the number of rooted spanning forests in the complete graph Kn. Some formulas were obtained for bipartite graphs [12], cyclic, star, line graphs [11] and some others [13]. In our previous paper [14],

The study of the second named author was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no. 0314—2019—0007).

we developed a new method for counting rooted spanning forests in circulant graphs. As for the number of unrooted forests, it has a much more complicated structure [15-17].

Another well-known invariant of a finite graph is Jacobian group (also known as the Picard group, critical group, sandpile group, dollar group). This concept was introduced independently by several authors [18-22], and [23]. In particular, the order of the Jacobian group coincides with the number of spanning trees for a graph. This fact is one of the reasons why interest in the Jacobian of a graph is manifested. However, the description of the Jacobian structure remains a difficult task and its structure is known only in several cases [18, 20, 24-27] and [28]. In this paper we intend to use the result [29] about the cokernel structure of Laplacian operator.

The paper is organized as follows.

Section 2 contains basic definitions and some known results on circulant graphs and circulant matrices. In Section 3, we describe general properties of cokernels for Z-linear operators represented by circulant matrices. The main result of Section 4 is Theorem 1 which asserts that the number of spanning trees in the cone over a graph G coincides with the number of rooted spanning forests in G. In Section 5, we introduce a notion of the forest group for a graph G defined as the cokernel of Z-linear operator I + L(G). Then the main result of the section (Theorem 2) states that Jacobian of the cone over a graph G is isomorphic to the forest group of G. Section 6 is devoted to description of Jacobian groups for the cone over a circulant graph and the cone over cobordism of two circulant graphs. Lastly, in Section 7, we use the obtained results to calculate Jacobian group and number of spanning trees for cones over some simple families of graphs.

2. Basic definitions and preliminary facts

Let G be a finite graph without loops. We denote the vertex and edge set of G by V(G) and E(G), respectively. Given u, v € V(G), we set auv to be equal to the number of edges between vertices u and v. The matrix A = A(G) = {auv}u,vev(g) is called the adjacency matrix of the graph G. The degree d(v) of a vertex v £ V(G)

is defined by d(v) = J2 auv. Let D = D(G) be the diagonal matrix of the

uEV (G)

size |V(G)| with dvv = d(v). The matrix L = L(G) = D(G) - A(G) is called the Laplacian matrix, or simply Laplacian, of the graph G.

Consider the Laplacian L(G) as a homomorphism Zn ^ Zn, where n is the number of vertices in G. The cokernel coker(L(G)) = Zn/im(L(G)) is an Abelian group. It can be uniquely represented in the form

coker(L(G)) ^ Zdl © Zd2 © • • • © Zdn,

where di satisfy the conditions di|di+i (1 < i < n).

Suppose that the graph G is connected, then the groups Zdl, Zd2,..., Zdn-1 are finite, and Zdn = Z. In this case, we define Jacobian of the graph G as

Jac(G) = Zdi © Zd2 © • • • © Zdn_i.

In other words, Jac(G) is isomorphic to the torsion subgroup of coker(L(G)).

Let si, s2, ■ ■ ■, Sk be integers such that 1 < si < s2 <•••< S& < ^. The graph Cn(si, ,..., sk) with n vertices 0,1, 2,..., n — 1 is called circulant graph if the vertex i, 0 < i < n—1 is adjacent to the vertices i±si, i±s2,..., i±sk (mod n). When Sk < f all vertices of a graph have even degree 2k. If n is even and Sk = then all vertices have odd degree 2k — 1. It is well known that the circulant Cn(s1, s2,..., sk) is connected if and only if gcd(s1; s2,..., sk, n) = 1.

We call an n x n matrix circulant, and denote it by circ(ao, a1;..., an-1) if it is of the form

circ(ao, &!,..., an-1)

! ao a1 a2 ... a«-1 \ ttn-1 ao a1 ... an-2

V a1 a2 a3 ... ao /

It easy to see that adjacency and Laplacian matrices of the circulant graph are circulant matrices. The converse is also true. If the Laplacian matrix of a graph is circulant then the graph is also circulant.

Recall [30] that the eigenvalues of matrix C = circ(ao, a1;..., an-1) are given by the following simple formulas Aj = P(e^), j = 0,1,... ,n — 1, where P(x) = ao + a1x + • • • + an-1xn-1 and en is an order n primitive root of the unity. Moreover, the circulant matrix C = P(T), where Tn = circ(0,1, 0,..., 0) is the matrix representation of the shift operator Tn : (xo, x1;..., xn-2, xn-1) ^ (x1, x2,..., xn-1, xo).

3. Cokernels of linear operators

Let P(z) be a bimonic integer Laurent polynomial. That is P(z) = zp+a1zp+1 +

• • • + as-1zp+s-1 + zp+s for some integers p, a1; a2,..., as-1 and some positive integer s. Introduce the following companion matrix for polynomial

O | Is-1

P{z): of-- .

— 1, —a1,..., —as-1

where 1s-1 is the identity (s — 1) x (s — 1) matrix. We note that jz/ is invertible and inverse matrix -1 is also integer matrix.

Let A = (aj, j £ Z) be a free Abelian group freely generated by elements oj,

j £ Z. Each element of A is a linear combination Cj aj with integer coefficients

j

Cj. Define the shift operator T : A ^ A as a Z-linear operator acting on generators of A by the rule T : aj ^ aj+1, j £ Z. Then T is an endomorphism of A.

Let P(z) be an arbitrary Laurent polynomial with integer coefficients, then A = P(T) is also an endomorphism of A. Since A is a linear combination of powers of T, the action of A on generators aj can be given by the infinite set of linear

transformations A : aj ^ a^j a^, j £ Z. Here all sums under consideration are

i

finite. We set flj = Y1 ai,jai. Then im A is a subgroup of A generated by flj, j £ Z.

i

Hence, coker A = A/ im A is an abstract Abelian group (xi, i £ Z | ai;jxi = 0, j £

Z) generated by xi, i G Z with the set of defining relations a^jxi = 0, j G Z.

i

Here Xj are images of aj under the canonical homomorphism A ^ A/ im A. Since T and A = P(T) commute, subgroup im A is invariant under the action of T. Hence, the actions of T and A are well defined on the factor group A/ im A and are given

by T : Xj ^ Xj+i and A : Xj ^ ai,j xi respectively.

i

This allows to present the group A/ im A as follows (xi, i G Z | P (T )xj = 0, j G Z). In a similar way, given a set P1(z), P2(z),..., Ps(z) of Laurent polynomials with integer coefficients, one can define the group (xi, i G Z | P1(T)xj = 0,P2(T)xj = 0,.. .,PS(T)xj = 0, j G Z).

We will use the following proposition. By I = In we denote the identity matrix of order n.

Proposition 1. Let P(z) be a bimonic Laurent polynomial with integer coefficients and of be a companion matrix of P(z). Consider L = P(Tn) : Zn ^ Zn as a Z-linear operator. Then

coker L = coker(^/n — I ).

Proof. Since the Laurent polynomial P(z) is bimonic, it can be represented in the form

P (z ) = zp + aizp+1 + • • • + as_izp+s-1 +

.p+s

where p, s, ai, a2,..., as-i are integers and s > 0. Then the corresponding companion matrix jzf is

0 | Js_i

- 1, -ai,..., -as_i

Let T be the shift operator defined by T : Xj ^ xj+i,j G Z. Note that for any j G Z the relations P(T)xj = 0 can be written as xj+s = — Xj — aixj+i — • • • —

as_ ixj+s_ i.

Let xj = (xj+i, xj+2,..., xj+s )4 be s-tuple of generators xj+i, xj+2,..., xj+s. Then the relation P(T)xj = 0 is equivalent to xj = jzfxj_i. Hence, we have xi = ¿2/xo and x_i = ¿2/-ixo, where xo = (xi, x2,..., xs)4. So, xj = J2fj xo for any j G Z. Conversely, the latter implies xj = jzfxj_i and, as a consequence, P(T)xj = 0 for all j G Z.

Consider coker A = A/ im A as an abstract Abelian group with the following representation (xj, i G Z | P (T )xj = 0, j G Z). We show that coker A = Zs. Indeed,

coker A = (x^ i G Z | P(T)x^- = 0, j G Z)

= (xj, i G Z | xj + aixj+i + • • • + as-ixj+s-i + xj+s = 0, j G Z) = (xj,i G Z | (xj+i,xj+2,. .. ,xj+s)t = ¿/(xj,xj+i, .. . ,xj+s_i)t, j G Z) = (xj, i G Z | (xj+i , xj+2, . . . , xj+s)* = J2/j (xi, x2, .. ., xs)4, j G Z) = (xi,x2,...,xs | —) = Zs.

Now, our aim is to find cokernel of L = P (Tn). In the operator notations

cokerL = (xj,i G Z | P(T)x^- = 0, (Tn — 1)x^-, j G Z).

We set B = Tn — 1 and note [29, Lemma 3.1] that

coker L = coker A/ im(B|coker a) = coker(B|coker A).

We describe the action of the endomorphism B|coker a on the coker A. Since the operators A = P(T) and T commute, the action T|coker a : xj ^ xj+1, j £ Z on the coker A is well defined. First of all, we describe the action of T|coker a on the set of generators x1, x2,..., xs. For any i = 1,..., s — 1, we have T|coker A(xi) = xi+1 and T|cokera(xs) = xs+1 = —x1 — a1x2 — ••• — as-2xs-1 — as-1xs. Hence, the action of T|coker a on the coker A is given by the matrix srf. Considering srf as an endomorphism of the coker A, we can write T|coker a = srf■ Then, B|coker a = — I and coker L = coker(^/n — I). □

4. The number of spanning trees of a cone over a graph

The joint of graphs G1 and G2 is called the graph G = G1 * G2, of order m + n, obtained from the disjoint union of G1 of order m, and G2 of order n, by additionally joining every vertex of G1 to every vertex of G2. If G2 = K1 (the one-vertex graph with no edges) we are going to call the graph G = G1 * K1 a cone over graph G1.

Let G be a graph on n vertices. We define xg(A) = det(A/n — L(G)) as the characteristic polynomial of matrix L(G), which is Laplcaian matrix of graph G. Its extended form is

Xg(a) = An + c„-1An-1 + ••• + C1 A.

The theorem by Kelmans and Chelnokov [10] states that the absolute value of coefficient Ck of xg(A) coincides with the number of rooted spanning k-forests in the graph G. So, the number of rooted spanning forests of the graph G can be found by the formula

f (G) = /1 + /2 + • • • + fn = |C1 — C2 + C3 — • • • + ( — 1)n-1|

= ( — 1)nXG( — 1) = det(/„ + L(G)). (1)

This result was independently obtained by many authors: see, for example, [31,11], and [14].

The main result of this section is the following theorem.

Theorem 1. The number of spanning trees t(G) in the graph G, which is a cone over a graph G, coincides with the number of rooted spanning forests f (G) in the graph G.

Proof. As a corollary of the well-known Matrix-Tree-Theorem [9], the number of spanning trees of graph G of order n, can be found by the next formula t(G) =

( _ 1) n — 1

—n—Xg(0)- According to the result by Kelmans [32,33] (see also [34, Corollary 3.7]), for characteristic polynomial of a joint of two graphs G1 and G2, of order m and n, we have

x(x — n — to)

Xg1*G2{X) = T-77-rXGi [x - n)XG2 {X - to .

(x — n)(x — m)

As a consequence, for a graph G = Gi * G2, where Gi = G and G2 = Ki, we obtain

( —1)" , T(G) = yXg(O)

n + 1 G

( — 1)" ' ( —1)m ( x(x — 1 — n) , , A'

n + 1 1 m +1\(x — 1)(x — n) /x=0

It is known that xk (x) = x, so we obtain

r(G) = ^ lim = (-1)»XG(-1).

v 7 n + 1 x^o (x — 1)(x — n) v ' AGV 7

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By making use of formula (1) we finish the proof. □

The following corollary gives a convenient way to calculate the complexity of cone over a graph.

Corollary 1. The number of spanning trees in the cone over a graph G is given by the formula |xg( — 1)|, where xg(x) is the Laplacian characteristic polynomial of G.

Remark to Theorem 1. There is a natural way to get a one-to-one correspondence between spanning trees in the cone G and rooted spanning forests in the graph G.

Indeed, consider G as a joint G * {vo} of G with one-vertex graph {vo}. Let t be a spanning tree in G. We note that vo is a vertex of t. Let vovj, j = 1, 2,..., k be all the edges of graph t coming from vertex vo. Then f = t O G is a spanning forest in G consisting of k trees ti, t2,..., tk chosen in such a way that Vj is a vertex of tj. So, the pairs (tj, Vj), j = 1, 2,..., k form a rooted spanning forest in G. In turn, if (tj, vj), j = 1, 2,..., k is a rooted spanning forest in G, then the graph t obtained as a union of edges vovj and trees tj, j = 1, 2,..., k is a spanning tree in G.

5. The Jacobian of a cone over a graph and the forest group

The aim of the current section is to prove the following theorem.

Theorem 2. Let G be a graph on n vertices. Then Jacobian of the cone over G is isomorphic to the cokernel of linear operator /„ + L(G). Here L(G) is the Laplacian matrix of G and /„ is the identity matrix of order n.

Proof. For any given graph G on n vertices denote by L(G) the Laplacian matrix of the graph G and by G a graph that is a cone over graph G. It easy to see that the Laplacian matrix of G can be represented in the following form

L(G) =( T ) , where /„ is an identity matrix of order n and 1n is a

V —1n ^n + L(G) /

vector (1,1,..., 1) of length n. To find the Jacobian of the G we use the following useful relation between the structure of the Laplacian matrix and the Jacobian of a graph [24].

Consider the Laplacian L(G) as a homomorphism Zn+1 ^ Zn+1, where n + 1 is the number of vertices in G. The cokernel coker(L(G)) = Zn+1 / im(L(G)) — is an Abelian group. Let

coker(L(G)) ^ Zdl © Zd2 © • • • © Zd„+1

be its Smith normal form satisfying the conditions dj|dj+i, 1 < i < n. As the graph G is connected, the groups Zd1, Zd2,..., Zdn — are finite, and Zdn+1 = Z. Here dj = where i = 1, 2,..., n +1, is the greatest common divisor of all i x i

minors of matrix L(G) and = 1. Then,

Jac(G) = Zdl © Zd2 © • • • © Zdn

is the Jacobian of the graph G.

To calculate the Smith normal form of a given matrix one can use the following basic operations to convert the matrix to a diagonal form.

1°. Add arbitrary integer linear combination of rows to any other row.

2°. Add arbitrary integer linear combination of columns to any other column.

3°. Interchange any two rows or columns.

The matrix T is the Laplacian matrix for graph G. So,

V -1n + L(G) J

the sum of all rows and the sum of all columns in this matrix are zero vectors. Adding all the other rows to the first row we get zero first row. Then we add to the first column the all remained columns to get zero first column. As such, one can easily check that up to operations 1° — 3° of the above matrix is equivalent 0 0

to matrix t , where 0 is a zero vector of length n. Therefore, all

\ 0 ^n + L(G) J

nonzero elements of the Smith normal form are completely defined by the matrix In + L(G). □

We note that matrix /n + L(G) is always non-singular. Also, coker(/n + L(G)) is an Abelian group whose size det(/n + L(G)) is equal to the number of rooted spanning forests in graph G. So, it is natural to call coker(/n + L(G)) as a forest group of G and denote it by F(G). Then, then main statement of Theorem 2 can be rephrased as follows:

The Jacobian of the cone over a graph G is isomorphic to its forest group F(G).

6. The Jacobian of a cone over a circulant graph

This section is devoted to Jacobians of specific classes of graphs. They are cones over three families of graphs. Namely, cones over circulant graphs G = Cn(si, s2,..., sk) with even valency, circulant graphs G = C2n(si, ..., sk, n) with odd valency and cobordisms of two circulant graphs. The typical example of the graph in the first family is a cone over the cyclic graph Cn = Cn(1) also known as Wheel graph W (n). The second family contains a cone over the Mobius ladder C2n(1, n). The third set is represented by a cone over the Prism graph Pr(n).

Denote by G the cone over graph G. Using the results of Section 5, we establish general structural theorems for Jac(G) or, equivalently, for the forest group F(G) = coker(1 + L(G)).

6.1. The forest group of a circulant graph of even valency. Consider a 2/s-valent circulant graph G = Cn(si, s2, ■ ■ •, Sk), where I < si <■■■< Sk < t;■ Its Laplacian has the form

L(G) = 2kJn — £ (T"1 + T„-Sl)

n n

l=i

where T" = circ(0,1, 0,..., 0) is the (n x n) circulant matrix representing the shift operator (xi, x2,..., xn-i, xn) ^ (x2, x3,..., xn, xi). Then the forest group coker(1 + L(G)) has the following presentation:

^x,, i G Z | (2k + 1)xj — ^(xj+Sl + xj_Sl) = 0, xj+" = xj, j G Zj .

By Proposition 1, we conclude that coker(1 + L(G)) is isomorphic to the coker(^/" —

k

I), where is a companion matrix of the Laurent polynomial 2k + 1 — (zsi + z-si).

i=i

Combine this observation with Theorem 2, we get the following result.

Theorem 3. Let G be a cone over the circulant graph G = C"(si, s2,..., sk),

where 1 < si < s2 < • • • < Sk < f. Then Jac(G) is isomorphic to coker(«#™ — I),

k

where j// is a companion matrix of the Laurent polynomial 2k + 1 — (zsi + z-si).

i=i

6.2. Forest group of circulant graph of odd valency. Consider a (2k + 1)-

valent circulant graph of the form

G = C2n(si, S2,..., sk, n), where 1 < si < S2 < • • • < sk < n. In this case, the Laplacian matrix of G is

k

(2k +1)/2„ — T2" — ^ (T2" + T2-„s0, j=i

where T2n = circ(0,1, 0,..., 0) is a (2n x 2n) circulant matrix. In order to get the forest group of G, we have to find coker(1 + L(G)). It can be viewed as an infinitely generated Abelian group satisfying the following set of relations:

^x,, i G Z | (2k + 2)xj — xj+" — ^(xj+s + xj-si) = 0, xj+2« = xj, j G Z^ .

By making use of the shift operator T : xj ^ xj+i, j G Z, we rewrite the last formula as

^x,, i G Z | ^2k + 2 — T" — ^(Tsi + T-si) j x^- = 0, (T2n — 1)xj- = 0, j G Z } .

We can increase the list of relations by ones that are linear combinations of elements of a given set. One of such combinations is

k

m

(T2n — 1) + B(T) ^2k + 2 — Tn — (TSl + T)J

= + 2 — (TSl + T— 1,

where

k

B(T) = 2k + 2 + Tn — ^ (TSl + T).

;=i

k

In turn, T2n — 1 is a linear combination of 2k + 2 — Tn — £ (TSl + T) and

;=i

/ k _ \2

f 2k + 2 — (TSl + T Sl) I — 1. So, it can be replaced by the latter expression in the group presentation. Hence, coker(/ + L(G)) admits the following presentation

i G Z | ^ ^2k + 2 - Tn - ^(TSl + Txj = 0, ^2k + 2 - ^(TSl + T

- 1J xj = 0, j G Z^.

By Proposition 1, the forest group coker(1 + L(G)) is isomorphic to

coker - (2k + 2)1 + ^(¿/Sl + ) j ,

where sf is a companion matrix of the Laurent polynomial

^2k + 2 - ^(zs + z-s - 1.

Applying Theorem 2, we rewrite the obtained result in the following form. Theorem 4. Let G be a cone over the circulant graph

G = C2n(si, S2, . . . , Sk, n), 1 < Si < S2 < • • • < Sk < n.

Then Jac(G) is isomorphic to coker- (2k + 2)1 + ^ (^Sj + sf-sj, where sf

is a companion matrix of the Laurent polynomial 2k + 2 - (zsj + z Sj m - 1.

V j=i J

6.3. The Jacobian of a cone over cobordism of two circulant graphs.

Consider two circulant graphs on n vertices, namely C1 = Gn(s1j1, s1j2,..., s1;k) and

2

C2 = C"(si i, si 2,..., si,i), with k and l jumps respectively. Then the cobordism of two circulant graphs Ci and C2 is a graph G which is obtained from Ci and C2 by connecting every ith vertex of Ci with the corresponding ith vertex of C2. The Laplacian matrix of G has the form

/ (2k + 1)1" — E (T„s1-r + T„-s1'r) —I" ^

r=i

—1" (2l + 1)1" — £ (T„s2'r + T„-S2'r) ,

\ r=i /

The simplest example is the prism graph Pr(n) that is a cobordism of two cyclic graphs C" (1).

The complexity and other spectral properties of the cobordism graph G were investigated in [35].

Denote by G the cone over graph G. The aim of this subsection is to find cokernel of 1 + L(G), that is Jac(G). To do this, we use two bi-infinite sequences xj, yj, j G Z. Then the cokernel of the linear operator 1 + L(G) is isomorphic to the group

/ k

< x,,y,,i G Z | (2k + 2)xj — ^(xj+si,r + xj-si,r) — yj = 0, xj+" — xj = 0,

\ r=i

(21 + 2)yj — ^(y j + S 2 + yj-S2,r ) — xj = 0 yj + " — yj = 0 j G Z] .

r=i /

We note that

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k

yj = (2k + 2)xj — ^ (xj + Sl,r + xj-Sl,r ) r=i

is an integer linear combinations of xj, j G Z. Equivalently, in the operator form,

/ k n

yj = I 2k + 2 - Y^(Tsi,r + T)

V r=1 y

Then the group above is isomorphic to

| ^ ^2k + 2 - Y(Tsi-r + T-si-r+ 2 - (Ts2-r + T-s2-r- lj xj = 0,

(Tn - l)xj, j G Zj.

By Proposition l and Theorem 2, we get the following result.

Theorem 5. Let G be a cone over the cobordism graph G. Then the Jacobian Jac(G) is isomorphic to the cokernel of the linear operator jz/n - I, where of is a companion matrix of the Laurent polynomial

^2k + 2 - ^(zsi-r + z-si-r) j + 2 - J2(zS2,r + z-s2-r) j - l.

Table 1

n Jac(M(n)) | Jac(M(n))|

Table 2

n Jac (Pr(n)) | Jac(Pr(n))|

7. Examples

1°. The wheel graph W(n). The graph W(n) is a cone over the cyclic graph Cn = Cn(1). By Theorem 1, the number of spanning trees t(W(n)) is equal

to the number of rooted spanning forests in C" counting earlier in [14]. Hence, T(W(n)) = 2T„(|) — 2. See also paper [2] for an alternating proof of this result.

By Theorem 3, Jacobian of the Wheel graph Jac(W(n)) is isomorphic to the cokernel of linear operator — /2, where jz/ = {{0,1}, { —1, 3}} is a companion matrix of the Laurent polynomial 3 — z — z-i. Direct calculations leads to the well-known result [24]: Jac(W(n)) is isomorphic to ZFn ©Z5Fn if n is even, and ZLn ©ZLn if n is odd, where F" and L" are the Fibonacci and Lucas numbers respectively.

2°. The cone oyer the Mobius ladder M(n). Recall that the Mobius ladder M(n) is the circulant graph C2"(1,n). By Theorems 1 and 2 from [14], the number of spanning trees in the cone over the Mobius ladder M(n) can be found as

t (M(n)) = 4(T"(3/2) — 1)(T"(5/2) + 1).

The Jacobian of the cone over the Mobius ladder M (n) is isomorphic to the cokernel of the linear operator jz/" — 4/4 +#/ + #/-i, where jz/ is a companion matrix of the Laurent polynomial (4 — z — z-i)2 — 1.

Numerical calculation of Jac(M(n)) and t(M(n)) = | Jac(M(n))| is given in Table 1.

3°. The cone oyer the prism graph Pr(n). This graph is a cone over cobordism of two cyclic graphs C". By arguments similar to those from the proof of Theorem 2 in [14], the number of spanning trees of the cone over the prism graph Pr(n) is given by the formula

t (Pr(n)) = 4(T"(3/2) — 1)(T"(5/2) — 1).

By Theorem 5, the Jacobian of the cone over the prism graph Pr(n) is isomorphic to the cokernel of the linear operator jzf" — /4, where jzf is a companion matrix of the Laurent polynomial (4 — z — z-i)2 — 1.

The numerical calculation of Jac(Pr(n)) and t(Pr(n)) = | Jac(Pr(n))| is given in Table 2.

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Received February 15, 2021 Revised March 12, 2021 Accepted May 26, 2021

L. A. Grunwald (ORCID 0000-0003-4622-5259) Sobolev Institute of Mathematics, 4 Koptyug Avenue, Novosibirsk 630090, Russia; Novosibirsk State University, 1 Pirogov Street, Novosibirsk 630090, Russia lfb_o@yahoo. co .uk

I. A. Mednykh (ORCID 0000-0001-7682-3917) Sobolev Institute of Mathematics, 4 Koptyug Avenue, Novosibirsk 630090, Russia; Novosibirsk State University, 1 Pirogov Street, Novosibirsk 630090, Russia ilyamednykh@mail.ru

ON THE JACOBIAN GROUP OF A CONE OVER A CIRCULANT GRAPH Текст научной статьи по специальности «Математика»

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