# Prevalent behavior and almost sure Poincaré-Bendixson Theorem
for smooth flows with invariant $k$-cones††thanks: Supported by NSF of China
No.11825106, 11971232 and 12090012.

Yi Wang  School of Mathematical ScienceUniversity of Science and Technology of
ChinaHefei, Anhui, 230026, P. R. China  Jinxiang Yao  School of Mathematical
ScienceUniversity of Science and Technology of ChinaHefei, Anhui, 230026, P.
R. China  Yufeng Zhang Corresponding author<EMAIL_ADDRESS>(Y. Zhang).
School of Mathematical ScienceUniversity of Science and Technology of
ChinaHefei, Anhui, 230026, P. R. China

###### Abstract

We investigate the global dynamics from a measure-theoretic perspective for
smooth flows with invariant cones of rank $k$. For such systems, it is shown
that prevalent (or equivalently, almost all) orbits will be pseudo-ordered or
convergent to equilibria. This reduces to Hirsch’s prevalent convergence
Theorem if the rank $k=1$; and implies an almost-sure Poincaré-Bendixson
Theorem for the case $k=2$. These results are then applied to obtain an almost
sure Poincaré-Bendixson theorem for high-dimensional differential equations.

##  1 Introduction and Main Results

In this article, we are interested in the global dynamics in measure-theoretic
sense for flows with invariant cones of rank $k$ (abbr. $k$-cone). Such a flow
is also called monotone with respect to $k$-cone $C$ (see Section 2). Here, a
$k$-cone $C\subset\mathbb{R}^{n}$ means a closed subset that contains a linear
$k$-dimensional subspace and no linear subspaces of higher dimension.

From a measure-theoretic (or probabilistic) perspective, prevalence is a quite
useful notion, introduced by Christensen [4] and Hunt et al.[17], that
describes properties of interest occurs for “almost surely”. For Euclidean
spaces, it is equivalent to notion of full Lebesgue measure (see Section 2).
Over decades since its development, prevalence have undergone extensive
investigations. We refer to [17, 21, 18, 5] (and references therein) for more
details.

In dynamical systems, prevalence is parallel to another classical notion
called genericity, which formulates in the topological sense the properties of
interest occurs residually (i.e., on a countable intersection of open dense
subsets in a Baire space). Many examples in dynamical systems are known to be
both generic in the topological sense and prevalent in measure-theoretic
sense. However, on the other hand, there are also many cases properties that
are generic are not prevalent, and vise-versa (see, e.g. [18, 21]).

We point out that a $k$-cone is not a standard cone defined in the literature,
but adopted from by Fusco-Oliva [9, 10] (see also Krasnoselśkii et al. [22]
for a Banach space). In particular, a convex cone $K$ (defined in the standard
sense) gives rise to a $1$-cone, $K\cup(-K)$. Therefore, the class of flows we
consider here naturally includes the classical monotone dynamical systems
since the ground-breaking work of M.W. Hirsch [11, 12, 13, 14] (see monographs
or surveys [1, 16, 34, 38, 39], we call it as classical monotone systems).
There is however an essential difference between a $k(\geq 2)$-cone and a
convex cone $K$, due to the lack of convexity in the former. The lack of
convexity requires substantially new ideas for exploring the implication of
“monotonicity” for the dynamics of the considered flows. Typical examples of
flows monotone with respect to $k(\geq 2)$-cones are the high-dimensional
competitive systems (see, for example [2, 3, 15, 20, 28, 29, 30, 19, 46]);
systems with quadratic cones and Lyapunov-like functions (see, for example,
[31, 41, 42]); and monotone cyclic feedback systems with negative feedback
([25, 26, 27]) arising from a wide range of neural and physiological control
systems, etc..

Classical monotone dynamical systems are abundant and important sources of
topological genericity and prevalence. For continuous-time systems, the
celebrated Hirsch’s generic convergence theorem first indicates that the
generic precompact orbit approaches the set of equilibria (see Poláčik [32,
33] and Smith-Thieme [40] for the improved version). Meanwhile, Enciso, Hirsch
and Smith [6] investigated the prevalent behavior and proved that the set of
points that converge to equilibria is prevalent. For discrete-time systems
(mappings), Poláčik-Tereščák [35] first proved that the generic convergence to
periodic orbits occurs provided that the mapping $F$ is of class
$C^{1,\alpha}$ (see Tereščák [45] and Wang-Yao [47] for $C^{1}$-smooth
systems). Very recently, the present authors [48] proved the set of points
that converge to periodic orbits is prevalent.

An insightful observation for classical strongly monotone systems exhibits a
kind of important typical nontrivial orbits, called pseudo-ordered orbits.
More generally, for a $k$-cone $C$, a nontrivial orbit
$O(x):=\\{\Phi_{t}(x):t\geq 0\\}$ is pseudo-ordered if $O(x)$ possesses one
pair of distinct points $\Phi_{t}(x)$, $\Phi_{s}(x)$ such that
$\Phi_{t}(x)-\Phi_{s}(x)\in C$. This is also first handled by Sanchez [36].
Very recently, Feng, Wang and Wu [8] proved that generic (open and dense)
orbits in $\mathbb{R}^{n}$ are either pseudo-ordered or convergent to
equilibria. This covers the Hirsch’s Generic Convergence Theorem in the case
$k=1$. Together with their previous work [7], a generic Poincaré-Bendixson
Theorem is thus obtained for the case $k=2$ (see [8, Theorem B]).

In the present paper, we focus on the global behavior of a flow in strongly
monotone with respect to $k(\geq 2)$-cone $C$ from a measure-theoretic
perspective. Before describing our main results and our approach, we formulate
the standard assumptions:

(H)

The flow $\Phi_{t}$ is $C^{1,\alpha}$-smooth and strongly monotone with
respect to a $k$-solid cone $C$ in $\mathbb{R}^{n}$. Moreover, the
$x$-derivative $D_{x}\Phi_{t}$ satisfies
$D_{x}\Phi_{t}(C\backslash\\{0\\})\subset{\rm Int}C$ for $t>0$.

We refer to Section 2 for more detailed definitions about strong monotonicity
and others. Throughout the paper, we always assume (H) holds. Let

$Q=\\{x\in\mathbb{R}^{n}:\text{ the orbit }O(x)\text{ is pseudo-ordered}\\}.$

$\textbf{Theorem A (Prevalent Dynamics Theorem)}.$ Let
$D\subset\mathbb{R}^{n}$ be an open bounded set such that the orbit set $O(D)$
of $D$ is bounded. Then the set $\\{x\in D:x\in Q\text{ or }\omega(x)\text{ is
a singleton}\\}$ is prevalent in $D$. Furthermore, the set has full Lebesgue
measure in $D$.

Theorem A concludes that, for almost every point in $D$, the orbit is either
pseudo-ordered or convergent to a single equilibrium. As we mentioned above,
the set

$D^{\prime}\triangleq\\{x\in D:x\in Q\text{ or }\omega(x)\text{ is a
singleton}\\}$

is also generic (i.e., contains an open and dense subset) in $D$ (c.f. [8,
Theorem A]). This entails that, for smooth flow strongly monotone with respect
to $k$-cone, the fact “convergence to an equilibrium or the orbit is pseudo-
ordered” is both prevalent in measure-theoretic sense and generic in the
topological sense.

Needless to say, if the rank $k=1$, Theorem A in conjunction with the Monotone
Convergence Criterion (see [38]) naturally reduces to the prevalent
convergence obtained by Enciso, Hirsch and Smith [6, Theorem 1] for
$\mathbb{R}^{n}$.

Moreover, if the rank $k=2$, we have the following:

$\textbf{Theorem B (Almost Sure Poincar\'{e}-Bendixson Theorem)}.$ Let all
hypotheses in Theorem A hold. Assume also that $k=2$ and $C$ is completed.
Then, for almost every point $x\in D$, the $\omega$-limit set $\omega(x)$
containing no equilibria is a single closed orbit.

As mentioned above, Feng et.al [8] obtained the so called “Generic Poincaré-
Bendixson Theorem”, that is, for generic (open and dense) points $x\in D$, the
$\omega$-limit set $\omega(x)$ containing no equilibria is a single closed
orbit. A drawback of topological genericity is that open dense subsets can be
arbitrarily small in terms of measure. However, Theorem B here exhibits that,
for smooth flow strongly monotone with respect to $2$-cone, the “Poincaré-
Bendixson Theorem” holds not only generically in the topological sense, but
also almost-surely in measure-theoretic sense.

Compared to the classical monotone dynamical flows, the generic/prevalent
behavior of the flows with invariant $k(\geq 2)$-cones is much more
complicated. Due to the loss of convexity, the “order”-relation defined by $C$
(see Section 2) is not a partial order, because it is neither antisymmetric
nor transitive. So, it requires substantially new techniques for exploring the
implication of monotonicity for the dynamics of the considered flows.

Our approach involves the ergodic argument using the $k$-exponential
separation and the associated $k$-Lyapunov exponent (that reduces to the first
Lyapunov exponent if $k=1$), which was first introduced in [8]. Plus, we
present a novel interesting lemma, called the Probe Lemma (see Section 3),
which plays a decisive role for the proof of the main results. Roughly
speaking, this lemma indicates that one can detect the pseudo-ordered orbits
surrounding the point $z\notin D^{\prime}$, by tracking the information
restricted on a $k$-dimensional probe $P$ passing through $z$ (see Lemma 3.1).
Based on the Probe Lemma, we succeeded in proving the main results in Section
4.

In section 5, our main results will be applied to an $n$-dimensional ODE
system with a quadratic cone and obtain the almost sure Poincaré-Bendixson
theorem in such high-dimensional system. R. A. Smith [41, 42] has ever
obtained a Poincaré-Bendixson theorem for such system under the assumption of
the existence of a quadratic Lyapunov function. The connections between
Smith’s results and the systems with invariant $2$-cones was observed by
Sanchez [36]. Feng et.al [8] proved a generic Poincaré-Bendixson Theorem even
if a quadratic Lyapunov function can not be constructed. Our Theorem B here
shows that for such system, the “Poincaré-Bendixson Theorem” holds not only
generically, but almost-surely in measure-theoretic sense.

##  2 Notations and Preliminary Results

In this section, we will fix some fundamental notations. Let $\mathbb{R}^{n}$
be the $n$-dimensional Euclidean space with a norm $\|\cdot\|$ and $\Phi(t,x)$
be the _flow_ on $\mathbb{R}^{n}$, which is a continuous map
$\Phi:\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ with:
$\Phi_{0}(x)=(x)$ and $\Phi_{t}\circ\Phi_{s}(x)=\Phi_{t+s}(x)$ for
$t,s\in\mathbb{R}$. We will henceforth continue to denote $\Phi(t,x)$ by
$\Phi_{t}(x)$. A _$C^{1,\alpha}$ -smooth_ flow $\Phi_{t}$ on $\mathbb{R}^{n}$
is a flow which $\Phi|_{\mathbb{R}\times\mathbb{R}^{n}}$ is a
$C^{1,\alpha}$-map (a $C^{1}$-map with a locally $\alpha$-Hölder derivative)
with $\alpha\in(0,1]$. We denote the _derivatives_ of $\Phi_{t}$ with respect
to $x$ at $(t,x)$ by $D_{x}\Phi_{t}$.

For a flow $\Phi(t,x)$ on $\mathbb{R}^{n}$, the _orbit of x_ is the set
$O(x)=\\{\Phi_{t}(x):t\in\mathbb{R}\\}$. The _$\omega$ -limit set_ of $x$ is
$\omega(x)=\cap_{s\geq 0}\overline{\cup_{t\geq s}\Phi_{t}(x)}$. We say that
the flow $\Phi_{t}$ is _$\omega$ -compact_ in a subset
$X\subset\mathbb{R}^{n}$ if $O(x)$ is relatively compact for each $x\in X$ and
$\bigcup_{x\in X}\omega(x)$ is relatively compact in $\mathbb{R}^{n}$. A point
$x$ is an _equilibrium_ if $O(x)=\\{x\\}$. The set of equilibria is denoted by
$\mathcal{E}$.

######  Definition 2.1.

Let $C\subset\mathbb{R}^{n}$ be a closed set. It is called a _k-cone_ of
$\mathbb{R}^{n}$ if it satisfies:

(i) For any $v\in C$ and $l\in\mathbb{R}$, $lv\in C$;

(ii) $\max\\{{\rm dim}\ W:W\subset C\text{ is a linear subspace}\\}=k$.
Moreover, the integer $k(\geq 1)$ is called the rank of $C$.

A $k$-cone $C\subset\mathbb{R}^{n}$ is called _$k$ -solid_ if there is a
$k$-dimensional linear subspace $W$ such that
$W\backslash\\{0\\}\subset\rm{Int}$$C$, and is called _complemented_ if there
is a $k$-codimensional space $H^{c}\subset\mathbb{R}^{n}$ such that $H^{c}\cap
C=\\{0\\}$. A pair $x,y\in\mathbb{R}^{n}$ are said to be _ordered_ if $x-y\in
C$, denoted by $x\sim y$; otherwise, $x$ and $y$ are called to be _unordered_.
We also write $x\approx y$ and call $x$ and $y$ are _strongly ordered_ if
$x-y\in\rm{Int}$$C$. Two sets $U,V\subset\mathbb{R}^{n}$ are said to be
_strongly ordered_ if $x-y\in C$ for any $x\in U$ and $y\in V$, denoted by
$U\approx V$.

Let $\Phi(t,x)$ denote a flow on $\mathbb{R}^{n}$. We call $\Phi(t,x)$
_monotone with respect to a k-solid cone C_ provided
$\Phi_{t}(x)\sim\Phi_{t}(y)$, whenever $x\sim y$ and $t\geq 0$; and _strongly
monotone with respect to a k-solid cone C_ provided
$\Phi_{t}(x)\approx\Phi_{t}(y)$, whenever $x\neq y$, $x\sim y$ and $t>0$. We
call a nontrivial orbit $O(x)$ _pseudo-ordered_ , if there are two distinct
points $\Phi_{t_{1}}(x)$, $\Phi_{t_{2}}(x)$ in $O(x)$ such that
$\Phi_{t_{1}}(x)\sim\Phi_{t_{2}}(x)$; otherwise, $O(x)$ is called _unordered_.

In the following, we give the definition of _prevalence_ (see, e.g. [18, 17,
6]). Let $\mathbb{B}$ be a Banach space. A Borel set $W\subset\mathbb{B}$ is
_shy_ if there exists a nonzero compactly supported Borel measure $\mu$ on
$\mathbb{B}$ such that $\mu(v+W)=0$ for every $v\in\mathbb{B}$. More
generally, a set is called _shy_ if it is contained in a shy Borel set. A set
is _prevalent_ if its complement is shy. Given $A\subset\mathbb{B}$, we say
that a set $W$ is _prevalent in_ $A$ if $A\setminus W$ is shy.

In general situation, shy sets have the following properties: (i) Every subset
of a shy set is shy; (ii) Every translation of a shy set is shy; (iii) No
nonempty open set is shy. (iv) Every countable union of shy sets is shy ([17,
18]). In finite-dimensional spaces, a set $M$ is shy if and only if it has
Lebesgue measure zero (see, e.g. [18, Proposition 2.5] or [17, Fact 6]).

##  3 The Probe Lemma

Before we focus on our main theorems, we need to present a technical lemma,
called the Probe Lemma, which turns out to be very crucial for our approach.
To simplify the notation, we write

$S=\\{x\in\mathbb{R}^{n}:\omega(x)\text{ is a singleton}\\}.$

Clearly, $S$ is a Borel set (see [6, lemma 9]).

We call a $k$-dim linear subspace $P\subset\mathbb{R}^{n}$ a _$k$ -probe_ if
$P\subset{\rm Int}C$ (motivated by [18]). Given any $x\in\mathbb{R}^{n}$, let
$P_{x}=x+P$. Clearly, $P_{x}=P_{y}$ if and only if $x-y\in P$. Let
$B(x,\varepsilon)$ be the $\varepsilon$-neighborhood of $x$. The set
$B^{P}(x,\varepsilon)=P_{x}\cap B(x,\varepsilon)$ is called the probe
$\varepsilon$-neighborhood of $x$.

######  Lemma 3.1 (Probe Lemma).

Let all hypotheses in Theorem A hold. For any $x\in D\setminus(Q\cup S)$ and
any $k$-probe $P$, there exists $\varepsilon>0$ such that
$B^{P}(x,\varepsilon)\backslash\\{x\\}\subset Q$.

######  Remark 3.2.

When the rank $k=1$, the classical Monotone Convergence Criterion (c.f. [38,
Theorem 1.2.1]) yields that $Q\subset S$ in $D$. Under such spacial
circumstances, the Probe Lemma naturally reduces to “For $x\in D\setminus S$,
there is a neighborhood $\mathcal{N}$ of $x$ such that, for any
$y\in\mathcal{N}$ with $y\sim x$, one has $y\in S$”. This fact is a part of
Sequential Limit Trichotomy (see, e.g. [38, Theorem 1.4.1(b) or Proposition
1.4.4(d)]) for classical monotone flows, or parallel to the Discontinuity
Principle (see, e.g. [43, Proposition 3.3]) for classical monotone discrete-
time systems.

Before proving the Probe Lemma, we are going to introduce some tools as
$k$-Exponential Separation and $k$-Lyapunov Exponents, which we inherit from
Feng, Wang and Wu [8] and play an important role in the following proof.

Let $K\subset\mathbb{R}^{n}$ be an invariant compact subset. Then,
$(\Phi_{t},D\Phi_{t})$ naturally defines a linear skew-product flow on
$K\times\mathbb{R}^{n}$. Let $\\{Z_{x}\\}_{x\in K}$ be a family of
$k$-dimensional subspaces of $\mathbb{R}^{n}$, and $G(k,\mathbb{R}^{n})$
consists of $k$-dim linear subspaces of $\mathbb{R}^{n}$ which is a complete
metric space by endowing the gap metric (see, e.g. [24, 23]). Then,
$K\times(Z_{x})$ is said to be a _$k$ -dimensional continuous vector bundle_
if the map $K\rightarrow G(k,\mathbb{R}^{n}):x\mapsto Z_{x}$ is continuous.
Also, we say the continuous vector bundle $K\times(Z_{x})$ is invariant with
respect to $(\Phi_{t},D\Phi_{t})$ if $D\Phi_{t}Z_{x}=Z_{\Phi_{t}x}$ for any
$x\in K$ and $t\geq 0$.

We call $(\Phi_{t},D\Phi_{t})$ admits a $k$_-exponential separation along_ $K$
(for short, $k$_-exponential separation_), if there are $k$-dimensional
continuous bundle $K\times(E_{x})$ and $(n-k)$-dimensional continuous bundle
$K\times(F_{x})$ such that

(i) $\mathbb{R}^{n}=E_{x}\oplus F_{x}$, for any $x\in K$;

(ii) $D_{x}\Phi_{t}E_{x}=E_{\Phi_{t}(x)}$, $D_{x}\Phi_{t}F_{x}\subset
F_{\Phi_{t}(x)}$ for any $x\in K$ and $t>0$;

(iii) there are constants $M>0$ and $0<\gamma<1$ such that

$\|D_{x}\Phi_{t}w\|\leq M\gamma^{t}\|D_{x}\Phi_{t}v\|$

for all $x\in K$, $w\in F_{x}\cap S$, $v\in E_{x}\cap S$ and $t\geq 0$, where
$S=\\{v\in\mathbb{R}^{n}:\|v\|=1\\}$.

In addition, if $C\subset\mathbb{R}^{n}$ is a $k$-solid cone and

(iv) $E_{x}\subset\text{Int}C\cup\\{0\\}$ and $F_{x}\cap C=\\{0\\}$ for any
$x\in K$,
then $(\Phi_{t},D\Phi_{t})$ is called to admit a $k$_-exponential separation
along_ $K$ _associated with_ $C$.

It is known that $(\Phi_{t},D\Phi_{t})$ admits a $k$-exponential separation
along $K$ associated with $C$ if the flow $\Phi_{t}$ satisfied our assumption
(H) (see, e.g. [44, Corollary 2.2 or Theorem 4.1] and [29, Proposition A.1]).

Let $K\subset\mathbb{R}^{n}$ be an invariant compact subset for $\Phi_{t}$,
and $(\Phi_{t},D\Phi_{t})$ admit a $k$-exponential separation

$K\times\mathbb{R}^{n}=E\oplus F,$

where $E=K\times(E_{x})$ and $F=K\times(F_{x})$. For each $x\in K$, we define
the $k$_-Lyapunov exponent_ as
$\lambda_{kx}=\limsup_{t\rightarrow\infty}\frac{\log
m(D_{x}\Phi_{t}|_{E_{x}})}{t},$ where $m(D_{x}\Phi_{t}|_{E_{x}})=\inf_{v\in
E_{x}\cap S}\|D_{x}\Phi_{t}v\|$ is the infimum norm of $D_{x}\Phi_{t}$
restricted to $E_{x}$. When $k=1$, $\lambda_{kx}$ naturally reduces to first
(or principal) Lyapunov exponent (see, e.g. [35]).

A _regular point_ is a point $x\in K$ for which
$\lambda_{kx}=\lim_{t\rightarrow\infty}\dfrac{\log
m(D_{x}\Phi_{t}|_{E_{x}})}{t}$.

######  Proposition 3.3.

For any $x\in D$, one of the alternatives must occur:

(i) $x\in Q\cup S$;

(ii) For any sequence $\\{x_{n}\\}_{n=1}^{\infty}\subset D\backslash\\{x\\}$
with $x_{n}\rightarrow x$ and $x_{n}\sim x$, there are two points
$p\in\omega(x)$, $q\in\omega_{c}(x)$ such that $p\sim q$, where
$\omega_{c}(x)=\cap_{n\geq 1}\overline{\cup_{m\geq n}\omega(x_{m})}$.

###### Proof.

Our proof is motivated by the approach in [8, Section 4]. Denote by
$\omega_{0}(x)$ the set of regular points on $\omega(x)$, i.e.,

$\omega_{0}(x)=\\{z\in\omega(x):z\text{ is a regular point}\\}.$

Then one of the following alternatives must occur:

(a) $\lambda_{kz}\leq 0$ for some point $z\in\omega_{0}(x)$;

(b) $\lambda_{kz}>0$ for any $z\in\omega(x)$;

(c) $\lambda_{kz}>0$ for any $z\in\omega_{0}(x)$; while
$\lambda_{k\widetilde{z}}\leq 0$ for some
$\widetilde{z}\in\omega(x)\backslash\omega_{0}(x)$.
We will discuss the three cases one by one.

If (a) holds, then it follows from [8, Theorem 4.2] that $x\in Q\cup S$. Thus,
item (i) holds.

If (b) holds, we choose any sequence $\\{x_{n}\\}_{n=1}^{\infty}\subset
D\backslash\\{x\\}$ satisfying $x_{n}\rightarrow x$ and $x_{n}\sim x$.
Clearly, $\omega_{c}(x)$ is nonempty, compact and invariant. By virtue of [8,
Lemma 4.4], there exists a constant $\delta>0$ such that

$\limsup_{t\rightarrow\infty}\|\Phi_{t}(x_{n})-\Phi_{t}(x)\|\geq\delta,\text{\
\ for each }n\geq 1.$

Consequently, for each $n\geq 1$, there are $p_{n}\in\omega(x)$ and
$q_{n}\in\omega(x_{n})$ such that $\|p_{n}-q_{n}\|\geq\delta$ and $p_{n}\sim
q_{n}$. Without loss of generality, one may assume that $p_{n}\rightarrow
p\in\omega(x)$ and $q_{n}\rightarrow q\in\omega_{c}(x)$, as
$n\rightarrow\infty$. Therefore, one has $p\sim q$ and $p\neq q$. Thus, item
(ii) holds.

Finally, if (c) holds, then for any $y\in\omega(x)$, we write
$v_{y}:=\frac{d}{dt}|_{t=0}\Phi_{t}(y)$ and let

$\lambda(z,v_{z})=\limsup_{t\rightarrow\infty}\frac{\log\|v_{\Phi_{t}(z)}\|}{t}\text{,\
\ for any }z\in\omega(x)\backslash\omega_{0}(x).$

Fix any $z\in\omega(x)\backslash\omega_{0}(x)$, clearly, $\lambda(z,v_{z})\leq
0$, since $\|v_{\Phi_{t}(z)}\|$ is bounded uniformly for $t\geq 0$.

When $\lambda(z,v_{z})=0$, we have $v_{z}\notin F_{z}$ (For otherwise, [8,
Lemma 3.6(i)] implies that $\lambda(z,v_{z})\leq\lambda_{kz}+\log(\gamma)<0$,
a contradiction). Therefore, by the $k$-exponential separation, there is $T>0$
such that $D_{z}\Phi_{t}(v_{z})\in\text{Int}C$ for any $t>T$, which implies
$z\in Q$. Noticing that $z\in\omega(x)$, we obtain $x\in Q$. Thus, item (i)
holds.

When $\lambda(z,v_{z})<0$, we have $v_{\Phi_{t}(z)}\rightarrow 0$ as
$t\to\infty$. This implies $\omega(z)$ only consists of equilibria. Thus,
$\omega(z)\subset\omega_{0}(x)$. Hence, due to (c), one has

$\lambda_{k\widetilde{z}}>0\text{,\ \ for any }\widetilde{z}\in\omega(z).$
(3.1)

Now, define $C_{x}=\\{y\in D:y\neq x\text{ and }y\sim x\\}$ and choose a
sequence $t_{n}\rightarrow\infty$ such that $\Phi_{t_{n}}(x)\rightarrow z$ as
$n\rightarrow\infty$.

If there exists $y\in C_{x}$ with a subsequence
$\\{t_{n_{i}}\\}_{i=1}^{\infty}$ of $\\{t_{n}\\}_{n=1}^{\infty}$, such that
$\Phi_{t_{n_{i}}}(y)\rightarrow z$, then [7, Lemma 4.3] (or [8, Lemma 2.4])
implies that $z\in Q\cup\mathcal{E}$. Recall that $z\notin\omega_{0}(x)$.
Then, we have $z\in Q$. Hence, we again obtain that $x\in Q$, which means item
(i) holds.

On the other hand, if for any $a\in C_{x}$, there is a subsequence
$t_{n_{j}}^{a}\rightarrow\infty$ of $\\{t_{n}\\}_{n=1}^{\infty}$ such that
$\Phi_{t_{n_{j}}^{a}}(a)\rightarrow z_{a}\neq z$ as $j\rightarrow\infty$.
Clearly, $z_{a}$ and $z$ are ordered for any $a\in C_{x}$. By virtue of (3.1),
we can again utilize [8, Lemma 4.4] (for $\omega(z)$) to obtain that there
exists $\delta>0$ such that

$\limsup_{t\rightarrow\infty}\|\Phi_{t}(z_{a})-\Phi_{t}(z)\|\geq\delta,\text{\
\ for any }a\in C_{x}.$ (3.2)

Take any sequence $\\{x_{k}\\}_{k=1}^{\infty}\subset D\backslash\\{x\\}$ with
$x_{k}\rightarrow x$ and $x_{k}\sim x$. For each $k$, we utilize (3.2) (hence,
for each $z_{x_{k}}$) to obtain that there are $p_{k}\in\omega(z)$ and
$q_{k}\in\omega(z_{x_{k}})$ such that $\|p_{k}-q_{k}\|\geq\delta$ and
$p_{k}\sim q_{k}$. Recall that $\Phi_{t_{n_{j}}^{x_{k}}}(x_{k})\rightarrow
z_{x_{k}}$ as $j\rightarrow\infty$, one has $z_{x_{k}}\in\omega(x_{k})$ and
hence $\omega(z_{x_{k}})\subset\omega(x_{k})$. Thus we obtain
$q_{k}\in\omega(x_{k})$. And also $p_{k}\in\omega(x)$ since
$\omega(z)\subset\omega(x)$. Without loss of generality, one may assume that
$p_{k}\rightarrow p\in\omega(x)$ and $q_{k}\rightarrow q\in\omega_{c}(x)$, as
$k\rightarrow\infty$. Therefore, one has $p\sim q$ and $p\neq q$, which means
item (ii) holds. Thus, we have completed the proof. ∎

Now, we are going to prove the Lemma 3.1.

_Proof of lemma 3.1_ (_Probe Lemma_). We prove this lemma by contradiction.
Suppose that there exists a sequence $x_{n}\rightarrow x$ with $x_{n}\neq x$,
such that $x_{n}\in P_{x}\setminus Q$ for any $n\in\mathbb{N}$.

Then, by recalling that $x\notin Q\cup S$, it follows from Proposition 3.3
that there are $p\in\omega(x)$ and $q\in\cap_{n\geq 1}\overline{\cup_{m\geq
n}\omega(x_{m})}$ such that $p\neq q$ and $p\sim q$. Since $\Phi$ is strongly
monotone, we may assume that $p\approx q$. Let $U$ and $V$ be the neighborhood
of $p$ and $q$, respectively, such that $U\approx V$. Choose two sequences
$s_{k}\rightarrow\infty$ and $n_{k}\rightarrow\infty$ such that
$\Phi_{s_{k}}(x_{n_{k}})\in V$ for all $k$ sufficiently large. Let also
$\tau>0$ be such that $\Phi_{\tau}(x)\in U$. Then, $\Phi_{\tau}(x_{n_{k}})\in
U$; and hence, $\Phi_{\tau}(x_{n_{k}})\approx\Phi_{s_{k}}(x_{n_{k}})$ for all
$k$ sufficiently large. As a consequence, one has $x_{n_{k}}\in Q$ for all $k$
sufficiently large, a contradiction. Thus, we have completed the proof.
$\square$

##  4 Proof of the main Theorems

In this section, we will prove our main Theorems. For this purpose, besides
the Lemma 3.1 (Probe Lemma), we still need the following lemma to verify the
shyness of a subset $W\subset\mathbb{R}^{n}$.

######  Lemma 4.1.

Let $W\subset\mathbb{R}^{n}$ be a Borel subset. Assume that there exists a
$k$-probe $P\subset\rm{int}$$C$ such that $W\cap P_{v}$ is at most countable
for any $v\in\mathbb{R}^{n}$. Then $W$ is shy.

###### Proof.

Fix the $k$-dimensional unit disc $D\subset P$. Define the measure on
$\mathbb{R}^{n}$ as

$\mu_{D}(A)\triangleq m(A\cap D),\text{\ \ for any
}A\in\mathfrak{B}(\mathbb{R}^{n}).$ (4.1)

Here $m$ is the Lebesgue measure on $P$, and $\mathfrak{B}(\mathbb{R}^{n})$
consists of Borel sets of $\mathbb{R}^{n}$ . Clearly, $\mu$ is a nonzero
compactly supported Borel measure on $\mathbb{R}^{n}$. Moreover, for any
$v\in\mathbb{R}^{n}$, one has

$(v+W)\cap D\subset(v+W)\cap P=v+(W\cap P_{-v}).$

Since $W\cap P_{-v}$ is at most countable, we have $m((v+W)\cap D)=0$. By
(4.1), this entails that $\mu_{D}(v+W)=0$. Therefore, $W$ is shy. ∎

Now we are ready to prove the main Theorems.

$\emph{Proof of Theorem A}.$ Let $N=D\backslash(Q\cup S)$. Then $N$ is a Borel
set, since $Q$ and $S$ are both Borel sets. Fix a probe $P\subset\rm{Int}$$C$.
In order to show that $N$ is shy, by Lemma 4.1, it suffices to show that
$N\cap P_{v}$ is at most countable, for any $v\in\mathbb{R}^{n}$.

To this end, fix any $x\in N\cap P_{v}$, then lemma 3.1 (Probe Lemma) implies
that there exists $\varepsilon>0$ such that
$B^{P}(x,\varepsilon)\backslash\\{x\\}\subset Q$. Consequently, $x$ can not be
an accumulation point in $N\cap P_{v}$. By the arbitrariness of $x$ and the
separability of $P_{v}$, we obtain that $N\cap P_{v}$ is at most countable.

Thus, we have proved that $N$ is shy. Hence, $Q\cup S$ is prevalent in $D$.
Finally, in $\mathbb{R}^{n}$, a subset is prevalent if and only if it has full
Lebesgue measure. Thus, $Q\cup S$ is full Lebesgue measure in $D$. We have
completed the proof. $\square$

$\emph{Proof of Theorem B}.$ By virtue of Theorem A, the subset $Q\cup S$ is
full Lebesgue measure in $D$. Recall that $k=2$. Then, for any $x\in
D\cap(Q\cup S)$, the fact $\omega(x)\cap E=\emptyset$ directly implies that
$x\in Q$. It then follows from Feng et al. [7, Theorem C] (see also
Sanchez[36, Theorem 1]) that $\omega(x)$ must be a periodic orbit. Thus, we
have completed the proof. $\square$

##  5 Applications

In this section, we will apply the obtained Almost Sure Poincaré-Bendixson
Theorem for high-dimensional autonomous systems to the ordinary differential
equations.

Consider a system of ODEs

$\displaystyle\dot{x}=F(x),\ \ \ x\in\mathbb{R}^{n},$ (5.1)

for which $F$ is a $C^{2}$-smooth vector filed defined in $\mathbb{R}^{n}$.
Let $\Phi_{t}$ be the flow generated by the system (5.1). We call the system
(5.1) is _dissipative_ if there exists a bounded set
$\Lambda\subset\mathbb{R}^{n}$ satisfying for each $x\in\mathbb{R}^{n}$ there
exists a constant $t_{0}>0$ such that $\Phi_{t}(x)\in\Lambda$ for all $t\geq
t_{0}$.

Let $C\subset\mathbb{R}^{n}$ be a $k$-cone. We call the system (5.1) is
$C$-cooperative if the fundamental solution matrix $X^{ij}(t)$ of the linear
system

$\dot{X}=Q^{ij}(t)X,\ \ \ U(0)=I$

satisfies

$Q^{ij}(t)(C\backslash\\{0\\})\subset\rm{Int}\emph{C},\ \text{for all}\ t>0,$
(5.2)

which $Q^{ij}(t)=\int_{0}^{1}DF(\tau\Phi_{t}(i)+(1-\tau)\Phi_{t}(j))d\tau.$

Specially, a system satisfying (5.2) is called strongly cooperative if $C$ is
a classical cone.

Assume now that $C\subset\mathbb{R}^{n}$ is a complemented $2$-solid cone.
Then, we give the following Almost Sure Poincaré-Bendixson Theorem for system
(5.1).

######  Theorem 5.1.

Assume that system (5.1) is dissipative and $C$-cooperative. Then, for almost
every point $x\in\mathbb{R}^{n}$, if the $\omega$-limit set $\omega(x)$
contains no equilibrium, then $\omega(x)$ is a periodic orbit.

###### Proof.

The flow $\Phi_{t}$ satisfies the assumption (H) if system (5.1) is
$C^{1,\alpha}$-smooth and $C$-cooperative by [36, Proposition 1].

Let $B_{i}=\\{x\in\mathbb{R}^{n}:\|x\|<i\\}$ for any integer $i\geq 1$. Since
the system (5.1) is dissipative, the orbit set $O(B_{i})$ of $B_{i}$ is
bounded. Let $\mu$ be the Lebesgue measure on $\mathbb{R}^{n}$. According to
the Almost Sure Poincaré-Bendixson Theorem, for each $B_{i}$, there is a
subset $A_{i}\subset B_{i}$ with $\mu(A_{i})=\mu(B_{i})$, such that for any
$x\in A_{i}$ the $\omega$-limit set containing no equilibria is a closed
orbit.

Now, let $A=\cup_{i=1}^{\infty}A_{i}$. Note that
$\mathbb{R}^{n}=\cup_{i=1}^{\infty}B_{i}$. We have $\mu(A^{c})=0$ as
$A^{c}=\cup_{i\geq 1}(B_{i}\backslash A_{i})$. Also, for any $x\in A$, the
$\omega$-limit set $\omega(x)$ containing no equilibria is a single closed
orbit. We have completed the proof. ∎

In the rest of this section, we specifically describe a quadratic cone which
is 2-solid and complemented. Let $Q$ be a constant non-singular $n\times n$
matrix which is real symmetric. If $Q$ has 2 negative eigenvalues and $(n-2)$
positive eigenvalues, then $C^{-}(Q)=\\{x\in\mathbb{R}^{n}:x^{\ast}Qx\leq
0\\}$ is a complemented 2-solid cone, which $x^{\ast}$ denote the transpose of
the vector $x$.

In [36, 37], Sanchez shows a sufficient conditions for $C^{-}(Q)$-cooperative
system. Let $\lambda:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a continuous
function such that the matrices

$Q\cdot DF(x)+(DF(x))^{\ast}\cdot Q+\lambda(x)Q<0,\ \text{for any}\
x\in\mathbb{R}^{n},$ (5.3)

where $(DF(x))^{\ast}$ is the transpose of the Jacobian $DF(x)$ and “$<$”
represents the usual order in the space of symmetric matrices. Then system
(5.1) is $C^{-}(Q)$-cooperative if it satisfies (5.3) (see more details in
Sanchez [36, Proposition 7]).

Along this, we obtain the following Almost Sure Poincaré-Bendixson Theorem for
high-dimensional systems (5.1) with a quadratic cone:

######  Corollary 5.2.

Assume that system _(_ 5.1 _)_ is dissipative and satisfies (5.3). Then, for
almost every point $x\in\mathbb{R}^{n}$, if the $\omega$-limit set $\omega(x)$
contains no equilibrium, then $\omega(x)$ is a periodic orbit.

######  Remark 5.3.

Recently, Feng et.al [8] proved a generic Poincaré-Bendixson Theorem for this
system (5.1) with the assumption (5.3). We here obtain an almost-sure
Poincaré-Bendixson Theorem for $n$-dimensional ODE system (5.1) with a
quadratic cone. Thus, we conclude, for $n$-dimensional ODE system (5.1) with a
quadratic cone, the “Poincaré-Bendixson Theorem” holds not only generically,
but also almost-surely in measure-theoretic sense.

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# The sandpile group of polygon rings and twisted polygon rings

Haiyan Chen  The School of Sciences
Jimei University
Fujian, China<EMAIL_ADDRESS>and  Bojan Mohar  Department of Mathematics
Simon Fraser University
Burnaby, BC, Canada<EMAIL_ADDRESS>
###### Abstract.

Let $C_{k_{1}},\ldots,C_{k_{n}}$ be cycles with $k_{i}\geq 2$ vertices ($1\leq
i\leq n$). By attaching these $n$ cycles together in a linear order, we obtain
a graph called a polygon chain. By attaching these $n$ cycles together in a
cyclic order, we obtain a graph, which is called a polygon ring if it can be
embedded on the plane; and called a twisted polygon ring if it can be embedded
on the Möbius band. It is known that the sandpile group of a polygon chain is
always cyclic. Furthermore, there exist edge generators. In this paper, we not
only show that the sandpile group of any (twisted) polygon ring can be
generated by at most three edges, but also give an explicit relation matrix
among these edges. So we obtain a uniform method to compute the sandpile group
of arbitrary (twisted) polygon rings, as well as the number of spanning trees
of (twisted) polygon rings. As an application, we compute the sandpile groups
of several infinite families of polygon rings, including some that have been
done before by ad hoc methods, such as, generalized wheel graphs, ladders and
Möbius ladders.

This work was done while the first author visited The Simon Fraser University.
The hospitality of the hosting institution is greatly acknowledged. The visit
was funded by the Fujian Provincial Education Department.

H. Y. Chen was supported by the National Natural Science Foundation of China
(grant numbers 11771181, 12071180).

B.M. was supported in part by the NSERC Discovery Grant R611450 (Canada), by
the Canada Research Chairs program, and by the Research Project J1-8130 of
ARRS (Slovenia).

On leave from IMFM, Department of Mathematics, University of Ljubljana.

##  1\. Introduction

The abelian sandpile models were firstly introduced in 1987 by three
physicists, Bak, Tang, and Wiesenfeld [4], who studied it mainly on the
integer grid graphs. In 1990, Dhar[20] generalized their model from grids to
arbitrary graphs. The abelian sandpile model of Dhar begins with a connected
graph $G=(V,E)$ and a distinguished vertex $q\in V$, called the _sink_. A
_configuration_ of $(G,q)$ is a vector $\vec{c}\in\mathbb{N}^{V-q}$. A non-
sink vertex $v$ is _stable_ if its degree satisfies $d(v)>\vec{c}(v)$;
otherwise it is _unstable_. Moreover, a configuration is _stable_ if every
vertex $v$ in $V-q$ is stable. _Toppling_ an unstable vertex $u\in V-q$ in
$\vec{c}$ is the operation performed by decreasing $\vec{c}(u)$ by the degree
$d(u)$, and for each neighbour $v$ of $u$ different from $q$, adding the
multiplicity $m(u,v)$ of the edge $uv$ to $\vec{c}(v)$. Starting from any
initial configuration $\vec{c}$, by performing a sequence of topplings, we
eventually arrive at a stable configuration. It is not hard to see that the
stabilization of an unstable configuration is unique [20, 9]. The stable
configuration associated to $\vec{c}$ will be denoted by $s(\vec{c})$. Now,
let $(\vec{c}+\vec{d})(u):=\vec{c}(u)+\vec{d}(u)$ for all $u\in V-q$ and
$\vec{c}\oplus\vec{d}:=s(\vec{c}+\vec{d})$. A configuration $\vec{c}$ is
_recurrent_ if it is stable and there exists a non-zero configuration
$\vec{r}$ such that $s(\vec{c}+\vec{r})=\vec{c}$. Dhar[20] proved that the
number of recurrent configurations is equal to the number of spanning trees of
$G$, and that the set of recurrent configurations with $\oplus$ as a binary
operation forms a finite abelian group, which is called the _sandpile group_
of $G$. Soon after that, it was found that the sandpile group is isomorphic to
a number of ‘classical’ abelian groups associated with graphs, such as the
group of components in Arithmetic Geometry [30], Jacobian group and Picard
group in Algebraic Geometry[5], the determinant group in lattice theory[2],
the critical group of a dollar game [8, 9].

As an abstract abelian group, the structure of the sandpile group is
independent of the choice of the sink $q$. We denote the sandpile group of $G$
by $S(G)$. The classification theorem for finite abelian groups asserts that
$S(G)$ has a direct sum decomposition

$S(G)=\mathbb{Z}_{d_{1}}\oplus\mathbb{Z}_{d_{2}}\oplus\cdots\oplus\mathbb{Z}_{d_{r}},$

where the integers $d_{1},\ldots,d_{r}$, called the _invariant factors_ of
$S(G)$, satisfy $d_{1}\geq 2$ and $d_{i}\mid d_{i+1}$ for $i=1,\dots,r-1$. The
number of factors $r$ is the minimum number of generators of $S(G)$, and is
denoted by $\mu(G)$.

The standard method of obtaining invariant factors of a finite abelian group
is first to choose a presentation of the group, and then compute the Smith
Normal From (SNF) of the matrix of relations. For the sandpile group, it is
well known that the Laplacian matrix is one of its relation matrices. By
computing the SNF of the Laplacian matrix, the sandpile groups for many
special families of regular or near regular graphs have been completely or
partially determined in the last twenty years, such as wheel graphs, complete
multipartite graphs, Cartesian product of complete graphs $K_{n}\square
K_{m}$, the ladder $K_{2}\square C_{n}$, the Möbius ladder graph, a Cayley
graph $D_{n}$ of dihedral group, the squared cycle $C^{2}_{n}$, the thick
cycle, etc.; see [14, 15, 26, 16, 19, 28, 27, 31, 34, 35, 24, 33, 1, 12, 7,
11, 22, 21, 32, 3, 23, 10, 29, 25, 36] and references therein.

However, for families of irregular graphs, the SNF of the Laplacian matrix is
not easy to get in general. So in [13], the authors of the present paper
determined the sandpile group of a family of irregular graphs by using the
relations determined by the cycles and cuts of the graph. Using this method,
it is easy to show that the sandpile group of a polygon chain is always
cyclic, which has been proved earlier in [6, 28]. Here, we shall follow this
idea to study the sandpile group of a (twisted) polygon ring. We first show
that the sandpile group of any (twisted) polygon ring can be generated by at
most three edges, and then we give an explicit relation matrix among these
edges. So the sandpile group of any (twisted) polygon ring is the direct sum
of at most three cyclic groups. Our result can be used for computing sandpile
groups of various infinite families of (twisted) polygon rings, by computing
the Smith Normal Form of the relation matrix. Examples include many known
families, such as generalized wheel graphs, ladder graphs, Möbius ladders,
etc.[15, 18, 19].

The paper is organized as follows. In Section 2, we cover preliminaries. In
Section 3, by using the structure properties of (twisted) polygon rings, we
not only show that the sandpile group of any polygon ring can be generated by
at most three edges (see Theorem 3.1), but also give a relation matrix for
these generators (see Theorem 3.2). This is the most valuable aspect of this
study, since these results provide a uniform method to compute the sandpile
group of any (twisted) polygon ring. As applications, we give in Section 4 the
explicit relation matrices for a special family of polygon rings. In Sections
5 and 6, we determine the structure of sandpile groups by computing the Smith
Normal Form of the matrices given in Section 4 for polygon rings and twisted
polygon rings, respectively.

##  2\. Preliminaries

Let $G=(V,E)$ be a connected graph with $n$ vertices and $m$ edges. Given an
arbitrary orientation $\mathcal{O}$ of $E$, and an oriented edge $e=(u,v)$,
$v$ is called the _head_ of $e$, denoted by $h(e)$, and $u$ is called the
__tail__ of $e$, denoted by $t(e)$. As the convention, if $e=(u,v),$ then
$-e=(v,u)$. Let $\mathbb{Z}V,\mathbb{Z}E$ denote the free abelian groups on
$V$ and $E$, respectively. More clearly, every element $x\in\mathbb{Z}V$ is
identified with the formal sum $\sum_{v\in V(G)}x(v)v$, where
$x(v)\in\mathbb{Z}$, and similarly for $y\in\mathbb{Z}E$.

Consider a cycle $C=v_{1}e_{1}v_{2}e_{2}\cdots v_{k}e_{k}v_{1}$ in the
undirected graph $G$. The sign of an edge $e$ in $C$ with respect to the
orientation $\mathcal{O}$ is $\sigma(e;C)=1$ if $C=vev$ is a loop at the
vertex $v$, and otherwise

$\sigma(e;C)=\left\\{\begin{aligned} 1,&\ \ \ \mbox{if}\ \ e\in C\ \mbox{and}\
\ t(e)=v_{i},h(e)=v_{i+1}\ \ \mbox{for some}\ \ i;\\\ -1,&\ \ \ \mbox{if}\ \
e\in C\ \ \mbox{and}\ \ t(e)=v_{i+1},h(e)=v_{i}\ \ \mbox{for some}\ \ i;\\\
0,&\ \ \ \mbox{otherwise\ ($e$ does not occur in $C$)}.\end{aligned}\right.$

Here we interpret indices module $k$, i.e., $v_{k+1}=v_{1}$. We then identify
$C$ with the formal sum $\sum_{e\in E}\sigma(e,C)e\in\mathbb{Z}E$.

For each nonempty $U\subset V$, the cut corresponding to $U$, denoted by
$c_{U}$, is the collection of edges with one end vertex in $U$ and the other
in the complement $\overline{U}$. For each $e\in E$, define the sign of $e$ in
$c_{U}$ with respect to the orientation $\mathcal{O}$ by

$\sigma(e;c_{U})=\left\\{\begin{aligned} 1,\ \ \ &\mbox{if}\ \ t(e)\in U\ \
\mbox{and}\ \ h(e)\in\overline{U};\\\ -1,\ \ \ &\mbox{if}\ \
t(e)\in\overline{U}\ \ \mbox{and}\ \ h(e)\in U;\\\ 0,\ \ \ &\mbox{otherwise\
($e$ does not occur in $c_{U}$)}.\end{aligned}\right.$

We then identify $c_{U}$ with the formal sum $\sum_{e\in
E}\sigma(e,c_{U})e\in\mathbb{Z}E$.

A vertex cut is the cut corresponding to a single vertex, $U=\\{v\\}$, and we
write $c_{v}$ for $c_{U}$ in this case.

######  Definition 2.1.

The (integral) _cycle space_ , $\mathcal{C}\subseteq\mathbb{Z}E$, is the
$\mathbb{Z}$-span of all cycles. The (integral) _cut space_ ,
$\mathcal{B}\subseteq\mathbb{Z}E$, is the $\mathbb{Z}$-span of all cuts.

Let $L(G)$ be the Laplacian matrix of $G$. It can be viewed as a (linear)
mapping $\mathcal{L}:$ $\mathbb{Z}V\rightarrow\mathbb{Z}V$. We also define a
mapping $\rho:$ $\mathbb{Z}V\rightarrow\mathbb{Z}$ as $\rho(\sum_{v\in
V}x(v)v)=\sum_{v\in V}x(v).$

Obviously, both $\mathcal{L}$ and $\rho$ are group homomorphisms. Then we have
the following well-known results.

######  Theorem 2.2.

Let $G=(V,E)$ be a graph. With the notation defined above, we have

$S(G)\cong\frac{\operatorname{Ker}(\rho)}{\operatorname{Im}(\mathcal{L})}\cong\frac{\mathbb{Z}E}{\mathcal{C}\oplus\mathcal{B}}\,,$

where $\operatorname{Ker}(.)$ and $\operatorname{Im}(.)$ denote the kernel and
the image of a mapping.

The middle presentation of the sandpile group in Theorem 2.2 is the well-known
Jacobian group (also known as Picard group) of the graph. The Jacobian
presentation has a natural set of generators for $S(G)$, for which the
Laplacian matrix $L(G)$ of $G$ is a relation matrix. For more details, see
[9].

Here we mainly focus on the second presentation. For any $e\in E$, let
$\delta_{e}=\sum_{f\in E}\delta_{e}(f)f\in\mathbb{Z}E$, where
$\delta_{e}(f)=1$ if $e=f$ and $0$ otherwise. Then the collection of all
$\delta_{e}$ is a natural set of generators of the sandpile group $S(G)$, and
the relations are given by the elements in $\mathcal{C}\oplus\mathcal{B}$. So
to find a relation matrix, we only need to find a basis of the cycle space
$\mathcal{C}$ and a basis of the cut space $\mathcal{B}$, respectively.

By considering an edge $e$ of the graph as the element $\delta_{e}$ of
$\mathbb{Z}E/(\mathcal{C}\oplus\mathcal{B})$, we consider the edges as
particular elements of the sandpile group. We say that a set $F$ of edges
_generates_ the sandpile group if $S(G)$ is generated by the set
$\\{\delta_{e}\mid e\in F\\}$.

Now we recall the definition and basic properties of the Smith Normal Form
(SNF) of an integer matrix. Let $M,N$ be two $n\times n$ integer matrices. The
two matrices are called _equivalent_ if there exist invertible integer
matrices $P$ and $Q$ (i.e., $|\det(P)|=|\det(Q)|=1$) such that $PMQ=N$. We
have the following well-known results.

######  Theorem 2.3.

(1) Each integer matrix $M$ with rank $r$, is equivalent to a diagonal matrix
$diag(d_{1},\ldots,d_{r},0,\ldots,0)$, where $d_{i}\,|\,d_{i+1}$,
$i=1,\ldots,r-1$, and all these integers are positive. Furthermore, the
$d_{i}$ are uniquely determined by

$d_{i}=\frac{\Delta_{i}}{\Delta_{i-1}},\ i=1,\ldots,r,$

where $\Delta_{i}$ (called $i$-th _determinant divisor_) equals the greatest
common divisor of all $i\times i$ minors of the matrix $M$ ($1\leq i\leq r$)
and $\Delta_{0}=1$.

(2) Let $A$ be a finite abelian group with presentation
$A=\\{g_{1},\ldots,g_{n}\ |\ \sum_{j=1}^{n}m_{ij}g_{j}=0,i=1,\ldots,n\\}$. If
$M=(m_{ij})$ is equivalent to the diagonal matrix
$diag(d_{1},\ldots,d_{r},0,\ldots,0)$ then

$A\cong\mathbb{Z}_{d_{1}}\oplus\cdots\oplus\mathbb{Z}_{d_{r}}.$

The diagonal matrix in Theorem 2.3 (1) is called the _Smith Normal Form_ of
$M$, and the integers $d_{i}$ are called _invariant factors_ of $M$. The
matrix $M$ related to a presentation of the abelian group $A$ in part (2) of
Theorem 2.3 is called the _relation matrix_ of $A$.

From (1) of the above theorem, we see that equivalent matrices have the same
invariant factors. And (2) says that the invariant factors of $A$ are just the
non-trivial invariant factors (those that are $\geq 2$) of its arbitrary
relation matrix. So, to determine the structure of a finite abelian group, it
is sufficient to find a set of generators and a complete set of relations
among them, then compute the Smith Normal Form of the corresponding relation
matrix. In this paper, we shall start from the natural set of generators
$\delta_{e}$ $(e\in E(G))$ to study the sandpile groups of polygon rings.

Let $a_{1},\dots,a_{n}$ and $b_{1},\dots,b_{n}$ be sequences of non-negative
integers, and let $a=\sum_{i=1}^{n}a_{i}$ and $b=\sum_{i=1}^{n}b_{i}$. Take
two paths, $Q^{1}=x_{0}x_{1}\dots x_{a}$ of length $a$ and
$Q^{2}=y_{0}y_{1}\dots y_{b}$ of length $b$. For $j=0,1,\dots n$, define the
vertex $v_{i}$ to be the vertex $x_{t}$, where $t=\sum_{i=1}^{j}a_{i}$.
Similarly, define the vertex $u_{i}$ to be the vertex $y_{t}$, where
$t=\sum_{i=1}^{j}b_{i}$. In particular, $v_{0}=x_{0}$ and $v_{n}=x_{a}$. We
now define the _polygon chain_ $G_{n}=G_{n}(a_{1}\dots a_{n};b_{1}\dots
b_{n})$ as the graph obtained from the union $Q^{1}\cup Q^{2}$ by adding edges
$v_{i}u_{i}$ for $i=0,1,\dots,n$.

Let $k_{i}=a_{i}+b_{i}+2$. The polygon chain defined above consists of a
sequence of polygons with $k_{1},\ldots,k_{n}$ sides. The first one of these
polygons is a cycle $C_{k_{1}}$ of length $k_{1}$ and the last one is the
cycle $C_{k_{n}}$. These two are called _end-polygons_. The edges $v_{0}u_{0}$
and $v_{n}u_{n}$ in the end-polygons are called _free edges_ of $G_{n}$ (see
Figure 1). By identifying $v_{0}$ with $v_{n}$, $u_{0}$ with $u_{n}$, and thus
identifying $v_{0}u_{0}$ with $v_{n}u_{n}$, we obtain a _polygon ring_ ,
denoted by $R_{n}$ (see Figure 2(a)). If we identify $v_{0}$ with $u_{n}$,
$u_{0}$ with $v_{n}$ we obtain a _twisted polygon ring_ $T_{n}$ (see Figure
3(a)).

Figure 1. Polygon chain $G_{5}(2\,1\,1\,0\,3;2\,0\,2\,0\,1)$ with its free
edges $e_{0},e_{5}$ drawn thick.

Figure 2. (a) The polygon ring $R_{5}(2\,1\,1\,0\,3;2\,0\,2\,0\,1)$ obtained
from $G_{5}$ in Figure 1 by identifying $e_{0}$ and $e_{5}$. (b) The oriented
polygon ring $R_{5}$.

Figure 3. (a) The twisted polygon ring $T_{5}(2\,1\,1\,0\,3;2\,0\,2\,0\,1)$
obtained from $G_{5}$ in Figure 1. (b) The oriented twisted polygon ring
$T_{5}$.

As we mentioned before, the sandpile group of any polygon chain is cyclic.
Moreover, the sandpile group $S(G_{n})$ only depends on
$(k_{1},\ldots,k_{n})$, and it is independent of the way that the polygons
stack together [6, 28]. We shall prove that the sandpile group of a (twisted)
polygon ring $R_{n}(T_{n})$ is always the direct sum of at most three cyclic
groups by showing it can be generated by at most three edges. However, it
should be noted that, unlike the polygon chain, the structure of the sandpile
group of a polygon ring depends on the way that the polygons stacked together.

Finally, recall that for any connected graph $G$, the cuts $c_{v},v\in V(G)$
generate the whole cut space of $G$, and deleting any one of them, the
remaining cuts are independent and still spanning. While for any plane graph,
the cycles corresponding to the faces generate the whole cycle space of $G$,
and deleting any one of them, the remaining cycles are independent and still
spanning. In the following, we write $e$ instead of $\delta_{e}$ for
simplicity. Given integers $a_{1},\ldots,a_{t}$, we write
$gcd(a_{1},\ldots,a_{t})$ for their greatest common divisor.

##  3\. The minimum number of generators of $S(R_{n})$ and $S(T_{n})$

In this section, we shall discuss the sandpile group of a polygon ring. Let
$n\geq 2$ and $a_{1},\dots,a_{n}$, $b_{1},\dots,b_{n}$, $a$, $b$, and
$k_{1},\ldots,k_{n}$ be integers as used in the definition of a polygon chain.
For the corresponding polygon chain $G_{n}$, let $e_{i}=u_{i}v_{i}$
$(i=0,1,\ldots,n)$ and the polygons be $C_{k_{i}}$, $i=1,\dots n$. Recall that
$k_{i}=a_{i}+b_{i}+2$ for $1\leq i\leq n$. Let $R_{n}=R_{n}(a_{1}\ldots
a_{n};b_{1}\ldots b_{n})$ be the polygon ring and $T_{n}=T_{n}(a_{1}\ldots
a_{n};b_{1}\ldots b_{n})$ be the corresponding twisted ring.

Now we fix an orientation $\mathcal{O}$ of $E(R_{n})$ as follows. We orient
all edges in $Q^{1}$ from $v_{0}$ towards $v_{n}$ and all edges in $Q^{2}$ in
the opposite way from $u_{n}$ towards $u_{0}$, while each edge
$e_{i}=(u_{i},v_{i})$ is oriented from $u_{i}$ to $v_{i}$, $1\leq i\leq n$.
See Figure 2(b) for an example. We use the same orientation for $T_{n}$, where
the identified free edge is oriented from $u_{0}$ to $v_{0}$.

Figure 4. The cycle $C_{k_{i}}(i<n)$ in an oriented polygon ring.

######  Theorem 3.1.

Let $R_{n}=R_{n}(a_{1}\ldots a_{n};b_{1}\ldots b_{n})$ and
$T_{n}=T_{n}(a_{1}\ldots a_{n};b_{1}\ldots b_{n})$. The minimum number of
generators of their sandpile group $S(R_{n})$ and $S(T_{n})$ (respectively) is
at most $3$.

###### Proof.

We first prove the result for $R_{n}$.

It is sufficient to show that there exist at most three edges such that every
edge $e\in E(R_{n})$ can be expressed by them (viewed as an element in
$S(R_{n})=\frac{\mathbb{Z}E}{\mathcal{C\oplus\mathcal{B}}}$). We shall use the
relations determined by cuts $c_{v},v\in V(R_{n})$ and polygons $C_{k_{i}}$
alternatively to prove the result. Let us recall the following simple fact,
which we shall use repeatedly without pointing it out every time. If $e$ and
$f$ are two edges incident with a vertex $v$ of degree $2$ and $h(e)=t(f)=v$,
then

$e\equiv f\pmod{c_{v}}.$

So bear in mind $f_{i}=f_{i,1}\equiv\cdots\equiv f_{i,a_{i}}$ and
$g_{i}=g_{i,1}\equiv\cdots\equiv g_{i,a_{i}}$.

Also note that, under the given orientation $\mathcal{O}$, the cycle
$C_{k_{i}}$ (see Figure 4) can be expressed as follows:

$C_{k_{i}}=e_{i-1}+f_{i,1}+\cdots+f_{i,a_{i}}-e_{i}+g_{i,1}+\cdots+g_{i,b_{i}},\qquad
i=1,\dots,n.$

That is

$C_{k_{i}}=e_{i-1}+a_{i}f_{i}-e_{i}+b_{i}g_{i},\qquad i=1,\dots,n.$ $None$

In (3.1), if $a_{i}=0$, then $f_{i}$ is not defined, and we treat the term
$a_{i}f_{i}$ as being zero. The same convention applies when $b_{i}=0$.

For clarity, we split the proof in four cases.

(i) There exists some $k$ such that $a_{k}>0,b_{k}>0$. Without loss of
generality, we may assume that $a_{1}>1$ and $b_{1}>1$. We will show that
every edge (as an element of $\mathbb{Z}E/(\mathcal{B}\oplus\mathcal{C})$) can
be expressed by the three edges $e_{1},f_{1},g_{1}$.

Now we start from the cuts $c_{v_{1}},c_{u_{1}}$ and the cycle $C_{k_{2}}$:

$\displaystyle c_{v_{1}}$ $\displaystyle=f_{1}+e_{1}-f_{2}\quad(\textrm{if
}a_{2}>0),$ $\displaystyle c_{u_{1}}$
$\displaystyle=g_{1}+e_{1}-g_{2}\quad(\textrm{if }b_{2}>0),$ $\displaystyle
C_{k_{2}}$ $\displaystyle=e_{1}+a_{2}f_{2}+b_{2}g_{2}-e_{2}.$

From here it is easy to see that $e_{2},f_{2},g_{2}$ can be expressed by
$e_{1},f_{1},g_{1}$. (We use the first and the second equality only when
$a_{2}$ and $b_{2}$ are nonzero, respectively.) Thus, the same holds for every
edge in $E(C_{k_{2}})$.

Now the proof proceeds by induction on $i$. Suppose that the edges in
$E_{i-1}=\cup_{j=2}^{i-1}E(C_{k_{j}})$ have been expressed by
$e_{1},f_{1},g_{1}$. If $a_{i}>0$ and $b_{i}>0$, then

$c_{v_{i-1}}=f_{i}-\sum_{e\in E_{i-1},h(e)=v_{i-1}}e\ \qquad\mbox{and}\ \qquad
c_{u_{i-1}}=g_{i}-\sum_{e\in E_{i-1},t(e)=u_{i-1}}e.$ $None$

By the induction, $f_{i}$ and $g_{i}$ can be expressed by $e_{1},f_{1},g_{1}$.
Then by (3.1), so does $e_{i}$. Thus, all edges in $C_{k_{i}}$ can be
expressed in this case. The proof is basically the same when at least one of
conditions $a_{i}=0$ or $b_{i}=0$ or $a_{i}=b_{i}=0$ holds. The details are
left to the reader.

(ii) $a_{1}=\cdots=a_{n}=b_{1}=\cdots=b_{n}=0$. In this case, by (3.1),
$e_{1}\equiv\cdots\equiv e_{n}$. So $\mu(R_{n})=1.$

(iii) $a_{1}=\cdots=a_{n}=0$ but there exists some $k$ with $b_{k}>0$, or
$b_{1}=\cdots=b_{n}=0$ but there exists some $k$ with $a_{k}>0$. Without loss
of generality, suppose $a_{1}>0$ and $b_{1}=\cdots=b_{n}=0$. By a similar
argument as in (i), it is easy to show that every edge $e\in E(R_{n})$ can be
expressed by $e_{1}$ and $f_{1}$. So in this case, $\mu(R_{n})\leq 2$.

(iv) $a_{i}\cdot b_{i}=0$ for $1\leq i\leq n$, but there exist $k$ and $l$
such that $a_{k}>0$ and $b_{l}>0$. Suppose $a_{1}>0$ and
$b_{1}=\cdots=b_{l-1}=0$, but $b_{l}>0$. In this case, every edge $e\in
E(R_{n})$ can be expressed by $e_{1}$, $f_{1}$ and $g_{l}$.

Combining the above together, we obtain the result for $R_{n}$.

Since the relations determined by cuts $c_{u_{i}},c_{v_{i}}$ ($i=1,\dots,n-1$)
and cycles $C_{k_{j}}$ ($j=2,\dots,n-1$) in $R_{n}$ and $T_{n}$ are the same,
it is easy to see that all edges in $T_{n}$ except $e_{n}$ can be expressed by
$e_{1},f_{1},g_{1}$ with the same procedure as in $R_{n}$. While $e_{n}$ is
determined by the relation

$e_{n}=-(e_{n-1}+a_{n}f_{n}+b_{n}g_{n}).$

So the result holds for $T_{n}$, too. ∎

Note that in the above expressing process, we only use the cycles $C_{k_{i}}$
($2\leq i\leq n$) and cuts $c_{v_{i}},c_{u_{i}},i=1,\ldots,n-1$. So there are
three independent elements that have not been used: two in the cycle space and
one in the cut space. They give a relation matrix for the three generators
$e_{1},f_{1},g_{1}$.

For $R_{n}$, we choose:

$C_{k_{1}}=a_{1}f_{1}+e_{n}+b_{1}g_{1}-e_{1},\
C^{1}=a_{1}f_{1}+a_{2}f_{2}+\cdots+a_{n}f_{n}\hbox{ and
}c_{v_{n}}=f_{n}+e_{n}-f_{1}.$ $None$

For $T_{n}$, we choose:

$C_{k_{1}}=a_{1}f_{1}+e_{n}+b_{1}g_{1}-e_{1},\
C^{1}=a_{1}f_{1}+a_{2}f_{2}+\cdots+a_{n}f_{n}+e_{n}\hbox{ and
}c_{v_{n}}=-g_{n}+e_{n}-f_{1}.$ $None$

Recall that the expressions for the edges are the same in $R_{n}$ and $T_{n}$
in terms of $e_{1},f_{1},g_{1}$, but different for $e_{n}$, which satisfies
$e_{n}(T_{n})=-e_{n}(R_{n})$ in the form. So suppose in $R_{n}$

$\left(\begin{array}[]{c}f_{n}\\\ e_{n}\\\
g_{n}\end{array}\right)=\left(\begin{array}[]{ccc}a_{11}&a_{12}&a_{13}\\\
a_{21}&a_{22}&a_{23}\\\
a_{31}&a_{32}&a_{33}\end{array}\right)\left(\begin{array}[]{c}f_{1}\\\
e_{1}\\\ g_{1}\end{array}\right).$

Then we have in $T_{n}$

$\left(\begin{array}[]{c}f_{n}\\\ e_{n}\\\
g_{n}\end{array}\right)=\left(\begin{array}[]{ccc}a_{11}&a_{12}&a_{13}\\\
-a_{21}&-a_{22}&-a_{23}\\\
a_{31}&a_{32}&a_{33}\end{array}\right)\left(\begin{array}[]{c}f_{1}\\\
e_{1}\\\ g_{1}\end{array}\right).$

By expressing

$a_{1}f_{1}+a_{2}f_{2}+\cdots+a_{n}f_{n}=c_{1}f_{1}+c_{2}e_{1}+c_{3}g_{3},$

we obtain the following result.

######  Theorem 3.2.

Let the integers $a_{ij}$ and $c_{i}$ $(1\leq i\leq 3,\ 1\leq j\leq 3)$ be as
introduced above. Then the following holds.

(1) For the polygon ring $R_{n}$, the generators $e_{1},f_{1}$ and $g_{1}$ of
its sandpile group $S(R_{n})$ have the relation matrix:

$M=\left(\begin{array}[]{ccc}a_{11}-a_{1}-1&a_{12}+1&a_{13}-b_{1}\\\
a_{21}+a_{1}&a_{22}-1&a_{23}+b_{1}\\\ c_{1}&c_{2}&c_{3}\end{array}\right).$

(2) For the twisted polygon ring $T_{n}$, the generators $e_{1},f_{1}$ and
$g_{1}$ of its sandpile group $S(T_{n})$ have the relation matrix:

$T=\left(\begin{array}[]{ccc}a_{31}+a_{1}+1&a_{32}-1&a_{33}+b_{1}\\\
a_{21}-a_{1}&a_{22}+1&a_{23}-b_{1}\\\
c_{1}-a_{1}&c_{2}+1&c_{3}-b_{1}\end{array}\right).$

###### Proof.

Given the notation, it is easy to see the coefficient matrices of (3.3) and
(3.4) are

$M_{1}=\left(\begin{array}[]{ccc}a_{11}+a_{21}-1&a_{12}+a_{22}&a_{13}+a_{23}\\\
a_{21}+a_{1}&a_{22}-1&a_{23}+b_{1}\\\
c_{1}&c_{2}&c_{3}\end{array}\right)\sim\left(\begin{array}[]{ccc}a_{11}-a_{1}-1&a_{12}+1&a_{13}-b_{1}\\\
a_{21}+a_{1}&a_{22}-1&a_{23}+b_{1}\\\ c_{1}&c_{2}&c_{3}\end{array}\right)=M$
$T_{1}=\left(\begin{array}[]{ccc}a_{31}+a_{21}+1&a_{32}+a_{22}&a_{33}+a_{23}\\\
a_{21}-a_{1}&a_{22}+1&a_{23}-b_{1}\\\
c_{1}-a_{21}&c_{2}-a_{22}&c_{3}-a_{23}\end{array}\right)\sim\left(\begin{array}[]{ccc}a_{31}+a_{1}+1&a_{32}-1&a_{33}+b_{1}\\\
a_{21}-a_{1}&a_{22}+1&a_{23}-b_{1}\\\
c_{1}-a_{1}&c_{2}+1&c_{3}-b_{1}\end{array}\right)=T.$

∎

Since the number of spanning trees $\tau(R_{n})=det(M)$ and
$\tau(T_{n})=det(T)$, we can get a general relation between $\tau(R_{n})$ and
$\tau(T_{n})$ from the above theorem. In the next section, we shall proceed to
give explicit relation matrices among generators for a family of special
(twisted) polygon rings: $R_{n}(a\ldots a;b\ldots b)$ and $T_{n}(a\ldots
a;b\ldots b)$, that is, $a_{1}=\cdots=a_{n}=a$ and $b_{1}=\cdots=b_{n}=b$. In
this case we will use a simplified notation $R_{n}(a,b)$ and $T_{n}(a,b)$ (see
Figure 4 for examples). Note that $R_{n}(0,0)$ is just the 2-vertex graph with
$n$ parallel edges. The graph $R_{n}(a,0)$ ($a>0$) is the _generalized wheel
graph_ , and $R_{n}(1,1)$ and $T_{n}(1,1)$ are the ladder and the Möbius
ladder graphs, respectively. The sandpile groups of these graphs were
discussed in many papers as we mentioned in the introduction.

Figure 5. (a) The polygon ring $R_{4}(2,0)$. (b) The polygon ring
$R_{4}(3,1)$. (c) The twisted polygon ring $T_{4}(3,1)$.

##  4\. The relation matrices of $R_{n}(a,b)$ and $T_{n}(a,b)$

In this section, we focus on the explicit expressions for the relation
matrices given in Theorem 3.2 for $R_{n}(a,b)$ and $T_{n}(a,b)$.

######  Lemma 4.1.

Let $a\geq b$ be non-negative integers with $a>0$. Let $(\tau_{n})_{n\geq 1}$
be the integer sequence satisfying $\tau_{n}=(a+b+2)\tau_{n-1}-\tau_{n-2}$
with initial values $\tau_{-1}=-1$ and $\tau_{0}=0$. Let

$A=\left(\begin{array}[]{ccc}1&1&0\\\ a&a+b+1&b\\\ 0&1&1\end{array}\right).$

Then

$A^{n}=\left(\begin{array}[]{ccc}\frac{b}{a+b}+\frac{a}{a+b}(\tau_{n}-\tau_{n-1})&\tau_{n}&-\frac{b}{a+b}+\frac{b}{a+b}(\tau_{n}-\tau_{n-1})\\\
a\tau_{n}&\tau_{n+1}-\tau_{n}&b\tau_{n}\\\
-\frac{a}{a+b}+\frac{a}{a+b}(\tau_{n}-\tau_{n-1})&\tau_{n}&\frac{a}{a+b}+\frac{b}{a+b}(\tau_{n}-\tau_{n-1})\end{array}\right).$

###### Proof.

We prove the result by induction on $n$. It is trivial to check that the
result holds for $n=0,1$. Suppose the result holds for $n=k$. Let us write
$\nu=\tau_{k}-\tau_{k-1}$ and $\nu^{\prime}=\tau_{k+1}-\tau_{k}$. Note that
$\nu^{\prime}=\nu+(a+b)\tau_{k}$. Now it is easy to verify the following:

$\displaystyle A^{k+1}$ $\displaystyle=A^{k}A$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{b}{a+b}+\frac{a}{a+b}\nu+a\tau_{k}&(a+b+2)\tau_{k}-\tau_{k-1}&-\frac{b}{a+b}+\frac{b}{a+b}\nu+b\tau_{k}\\\
a\tau_{k+1}&(a+b+1)\tau_{k+1}-\tau_{k}&b\tau_{k+1}\\\
-\frac{a}{a+b}+\frac{a}{a+b}\nu+a\tau_{k}&(a+b+2)\tau_{k}-\tau_{k-1}&\frac{a}{a+b}+\frac{b}{a+b}\nu+b\tau_{k}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{b}{a+b}+\frac{a}{a+b}\nu^{\prime}&\tau_{k+1}&-\frac{b}{a+b}+\frac{b}{a+b}\nu^{\prime}\\\
a\tau_{k+1}&\tau_{k+2}-\tau_{k+1}&b\tau_{k+1}\\\
-\frac{a}{a+b}+\frac{a}{a+b}\nu^{\prime}&\tau_{k+1}&\frac{a}{a+b}+\frac{b}{a+b}\nu^{\prime}\end{array}\right).$

This completes the proof. ∎

Remark 1. Let $G_{n}$ be the polygon chain $G_{n}(s,\ldots,s)$, where
$s=a+b+2$. Then it is easy to see that $\tau_{n}$ in Lemma 4.1 is equal to the
number of spanning trees of $G_{n-1}$ ($n>0$) since they satisfy the same
linear recurrence relation with the same initial values. Solving the linear
recurrence relation, we have

$\tau_{n}=\frac{1}{\sqrt{s^{2}-4}}\left(\lambda_{1}^{n}-\lambda_{2}^{n}\right),$
$None$

where

$\lambda_{1}=\frac{s+\sqrt{s^{2}-4}}{2},\ \ \
\lambda_{2}=\frac{s-\sqrt{s^{2}-4}}{2}$

are the roots of $x^{2}-sx+1=0$. From (4.1), it is easy to deduce that

$\sum_{i=0}^{n-1}\tau_{i}=\frac{\tau_{n}-\tau_{n-1}-1}{a+b}.$ $None$
$\tau_{n-1}^{2}-\tau_{n}\tau_{n-2}=1.$ $None$

In the following, for simplicity, we always assume that $a\geq b$ and $n\geq
2$ for $R_{n}(a,b)$ and for $T_{n}(a,b)$.

######  Theorem 4.2.

Let $e_{i},f_{i}$ and $g_{i}$ be as given in Theorem 3.1. Then the following
holds.

(1) For the polygon ring $R_{n}(a,b)$ with $a\geq b>0$, the generators
$e_{1},f_{1}$ and $g_{1}$ have the relation matrix:

$M=\left(\begin{array}[]{ccc}\frac{a}{a+b}(\tau_{n}-\tau_{n-1}-1)&\tau_{n}&\frac{b}{a+b}(\tau_{n}-\tau_{n-1}-1)\\\
a(\tau_{n-1}+1)&\tau_{n}-\tau_{n-1}-1&b(\tau_{n-1}+1)\\\
\frac{a}{a+b}(nb+a(\tau_{n-1}+1))&\frac{a}{a+b}(\tau_{n}-\tau_{n-1}-1)&\frac{b}{a+b}(-na+a(\tau_{n-1}+1))\end{array}\right).$

(2) For the polygon ring $R_{n}(a,0)$ with $a>0$, the generators $e_{1},f_{1}$
have the relation matrix:

$N=\left(\begin{array}[]{cc}\tau_{n}-\tau_{n-1}-1&\tau_{n}\\\
a(\tau_{n-1}+1)&\tau_{n}-\tau_{n-1}-1\\\ \end{array}\right).$

(3) For the twisted polygon ring $T_{n}(a,b)$ with $a\geq b>0$, the generators
$e_{1},f_{1}$ and $g_{1}$ have the relation matrix:

$T=\left(\begin{array}[]{ccc}\frac{a}{a+b}(\tau_{n}-\tau_{n-1}-1)+1&\tau_{n}&\frac{b}{a+b}(\tau_{n}-\tau_{n-1}-1)+1\\\
a(\tau_{n-1}-1)&\tau_{n}-\tau_{n-1}+1&b(\tau_{n-1}-1)\\\
\frac{a}{a+b}(nb+a-b\tau_{n-1})&-\frac{b}{a+b}(\tau_{n}-\tau_{n-1}-1)-1&\frac{b}{a+b}(-na+a-b\tau_{n-1})\end{array}\right).$

###### Proof.

Let us first consider the polygon ring $R_{n}(a,b)$. By Theorem 3.1, we know
that $e_{1},f_{1}$ and $g_{1}$ generate the sandpile group $S(R_{n})$.
Moreover, by (3.1) and (3.2), for $i>1$, we have

$\displaystyle f_{i}$ $\displaystyle=f_{i-1}+e_{i-1},$ $\displaystyle g_{i}$
$\displaystyle=e_{i-1}+g_{i-1},$ $\displaystyle e_{i}$
$\displaystyle=e_{i-1}+af_{i}+bg_{i}=af_{i-1}+(a+b+1)e_{i-1}+bg_{i-1}.$

Let $\alpha_{i}=(f_{i},e_{i},g_{i})^{t}$. Then the above relation can be
expressed as $\alpha_{i}=A\alpha_{i-1}$, where $A$ is the matrix in Lemma 4.1.
So

$\alpha_{i}=A^{i-1}\alpha_{1},\ \ i\geq 1.$

Furthermore, by (4,1) and (4.2), we have

$\displaystyle C^{1}$ $\displaystyle=a(f_{1}+f_{2}+\cdots+f_{n})$
$\displaystyle=\frac{a}{a+b}[(nb+a(\tau_{n-1}+1))f_{1}+(\tau_{n}-\tau_{n-1}-1)e_{1}+(-nb+b(\tau_{n-1}+1))g_{1}].$

Then the matrices $M_{1}$ and $T_{1}$ in Theorem 3.2 have the form $M$ and $T$
in (1) and (3), respectively.

Part (2) follows from (1) by letting $b=0$ and deleting the third relation. ∎

From the above theorem, we immediately derive the following corollary.

######  Corollary 4.3.

Let $R_{n}=R_{n}(a,b)$ be a polygon ring, and let $T_{n}=T_{n}(a,b)$ be a
twisted polygon ring. The number of spanning trees of these graphs is given by
the formula:

$\tau(R_{n})=|S(R_{n})|=\begin{cases}\frac{nab}{a+b}(\tau_{n+1}-\tau_{n-1}-2)=\frac{nab}{a+b}(\lambda_{1}^{n}+\lambda_{2}^{n}-2),&\mbox{if
}a>0,\ b>0\\\
\tau_{n+1}-\tau_{n-1}-2=\lambda_{1}^{n}+\lambda_{2}^{n}-2,&\mbox{if }a>0,\
b=0\end{cases}$

and

$\tau(T_{n})=|S(T_{n})|=\frac{nab}{a+b}(\tau_{n+1}-\tau_{n-1}-2)+\frac{4nab}{a+b}+\frac{(a-b)^{2}}{a+b}\tau_{n}.$

###### Proof.

We only give the proof for $R_{n}(a,b)$ with $a>0$, $b>0$. The other cases can
be deduced similarly. By Theorem 4.2, $M$ is a relation matrix of $S(R_{n})$
for $a>0$, $b>0$. So

$\displaystyle\tau(R_{n})$ $\displaystyle=|S(R_{n})|=|det(M)|$
$\displaystyle=\left|\begin{array}[]{ccc}\frac{a}{a+b}(\tau_{n}-\tau_{n-1}-1)&\tau_{n}&\frac{b}{a+b}(\tau_{n}-\tau_{n-1}-1)\\\
a(\tau_{n-}+1)&\tau_{n}-\tau_{n-1}-1&b(\tau_{n-1}+1)\\\
\frac{a}{a+b}(nb+a(\tau_{n-1}+1))&\frac{a}{a+b}(\tau_{n}-\tau_{n-1}-1)&\frac{b}{a+b}(-na+a(\tau_{n-1}+1))\end{array}\right|$
$\displaystyle=\left|\begin{array}[]{ccc}\frac{a}{a+b}(\tau_{n}-\tau_{n-1}-1)&\tau_{n}&0\\\
a(\tau_{n-1}+1)&\tau_{n}-\tau_{n-1}-1&0\\\
\frac{a}{a+b}(nb+a(\tau_{n-1}+1))&\frac{a}{a+b}(\tau_{n}-\tau_{n-1}-1)&-nb\end{array}\right|$
$\displaystyle=-nab\left(\frac{1}{a+b}(\tau_{n}-\tau_{n-1}-1)^{2}-\tau_{n}(\tau_{n-1}+1)\right)$
$\displaystyle=\frac{nab}{a+b}(\tau_{n+1}-\tau_{n-1}-2)$
$\displaystyle=\frac{nab}{a+b}(\lambda_{1}^{n}+\lambda_{2}^{n}-2)$

where the next to the last equality is obtained from the recurrence relation
and (4.3). ∎

Remark 2. Recall that $\tau_{n}$ is equal to the number of spanning trees of
polygon chain $G_{n-1}=G_{n-1}(a+b+2,\ldots,a+b+2)$. As a by-product, we have
the following elegant relations:

$\tau(R_{n}(a,b)=\left\\{\begin{aligned}
&\frac{nab}{a+b}(\tau(G_{n})-\tau(G_{n-2})-2),\ \ \mbox{if}\ \ a>0,\ b>0;\\\
&\tau(G_{n})-\tau(G_{n-2})-2,\ \ \mbox{if}\ \ a>0,\ b=0;\end{aligned}\right.$
$\tau(T_{n}(a,b))=\tau(R_{n}(a,b))+\frac{4nab}{a+b}+\frac{(a-b)^{2}}{a+b}\tau_{n}.$

Specially,

$\tau(T_{n}(a,a))=\tau(R_{n}(a,a))+2na.$

In the next two sections, we shall determine the sandpile group of
$R_{n}(a,b)$ and $T_{n}(a,b)$ by computing the Smith Normal Forms of the
matrices $M$ and $T$.

##  5\. The sandpile group of $R_{n}(a,b)$

In this section, we shall focus on the Smith Normal Form of $M$ in Theorem
4.2. First we give some divisibility properties of $\tau_{n}$. Recall that
$\tau_{n}$ is an integer sequence satisfying
$\tau_{n}=(a+b+2)\tau_{n-1}-\tau_{n-2}$ with initial values $\tau_{-1}=-1$ and
$\tau_{0}=0$. For clarity, we define three new integer sequences
$\beta_{n},\gamma_{n},\delta_{n}$ that satisfy the same linear recurrence
relation as $\tau_{n}$ with initial values

$\displaystyle\beta_{0}=2,\ \beta_{1}=a+b+2;$ $\displaystyle\gamma_{0}=-1,\
\gamma_{1}=1;$ $\displaystyle\delta_{0}=1,\ \delta_{1}=a+b+1.$

Let $s=a+b+2$ and $t=\sqrt{s^{2}-4}$. Then the two roots of $x^{2}-sx+1$ are

$\lambda_{1}=\frac{s+t}{2},\ \ \lambda_{2}=\frac{s-t}{2}$

and

$\displaystyle\tau_{n}=\frac{1}{t}(\lambda_{1}^{n}-\lambda_{2}^{n}),$
$\displaystyle\beta_{n}=\lambda_{1}^{n}+\lambda_{2}^{n},$
$\displaystyle\gamma_{n}=\frac{-(a+b)+t}{2(a+b)}\lambda_{1}^{n}-\frac{(a+b)+t}{2(a+b)}\lambda_{2}^{n},$
$\displaystyle\delta_{n}=\frac{a+b+4+t}{2(a+b+4)}\lambda_{1}^{n}+\frac{a+b+4-t}{2(a+b+4)}\lambda_{2}^{n}.$

Furthermore, we write

$\rho_{n}=\frac{\tau_{n}-\tau_{n-1}-1}{a+b},\ \
s_{n}=\frac{\tau_{n+1}-\tau_{n-1}-2}{a+b}.$

Then we have the following results.

######  Lemma 5.1.

(1) If $n$ is even, then

$\displaystyle\tau_{n}=\tau_{\frac{n}{2}}\beta_{\frac{n}{2}};$
$\displaystyle\tau_{n-1}+1=\tau_{\frac{n}{2}}\beta_{\frac{n}{2}-1};$
$\displaystyle\tau_{n-1}-1=\tau_{\frac{n}{2}-1}\beta_{\frac{n}{2}};$
$\displaystyle\rho_{n}=\tau_{\frac{n}{2}}\gamma_{\frac{n}{2}};$ $\displaystyle
s_{n}=(a+b+4)\tau_{\frac{n}{2}}^{2}.$

(2) If $n$ is odd, then

$\displaystyle\tau_{n}=\gamma_{\frac{n+1}{2}}\delta_{\frac{n-1}{2}};$
$\displaystyle\tau_{n-1}+1=\gamma_{\frac{n+1}{2}}\delta_{\frac{n-3}{2}};$
$\displaystyle\tau_{n-1}-1=\gamma_{\frac{n-1}{2}}\delta_{\frac{n-1}{2}};$
$\displaystyle\rho_{n}=\tau_{\frac{n-1}{2}}\gamma_{\frac{n+1}{2}};$
$\displaystyle s_{n}=\gamma_{\frac{n+1}{2}}^{2}.$

###### Proof.

Using the facts $\lambda_{1}\lambda_{2}=1$, $\lambda_{1}+\lambda_{2}=s$,
$\lambda_{1}-\lambda_{2}=t$, the identities are easy to check. ∎

######  Lemma 5.2.

For $\tau_{n},\beta_{n},\gamma_{n},\delta_{n}$ as defined above, we have

$gcd(\tau_{n},\tau_{n-1})=gcd(\gamma_{n},\gamma_{n-1})=gcd(\delta_{n},\delta_{n-1})=1$

and

$gcd(\beta_{n},\beta_{n-1})=\begin{cases}1,&\mbox{if }a+b\mbox{ is odd},\\\
2,&\mbox{if }a+b\mbox{ is even}.\end{cases}$

###### Proof.

We only need to notice that, for any sequence $x_{n}$ satisfying
$x_{n}=(a+b+2)x_{n-1}-x_{n-2}$,

$gcd(x_{n},x_{n-1})=gcd(x_{n-1},x_{n-2})=\cdots=gcd(x_{1},x_{0}).$

Then the results follow directly. ∎

######  Theorem 5.3.

Let $a>0$ be an integer. The sandpile group of $R_{n}(a,0)$ is equal to

$S(R_{n}(a,0))=\left\\{\begin{aligned}
\mathbb{Z}_{\tau_{n}}\oplus\mathbb{Z}_{a(a+4)\tau_{n}},\ \ \ &\mbox{if $n$ is
even and $a$ is odd};\\\
\mathbb{Z}_{2\tau_{n}}\oplus\mathbb{Z}_{\frac{a(a+4)}{2}}\tau_{n},\ \ \
&\mbox{if $n$ is even and $a$ is even};\\\
\mathbb{Z}_{\gamma_{m+1}}\oplus\mathbb{Z}_{a\gamma_{m+1}},\ \ \ &\mbox{if
$n=2m+1$ is odd}.\end{aligned}\right.$

###### Proof.

By Theorem 4.2, the relation matrix of $R_{n}(a,0)$ is

$N=\left(\begin{array}[]{cc}\tau_{n}-\tau_{n-1}-1&\tau_{n}\\\
a(\tau_{n-1}+1)&\tau_{n}-\tau_{n-1}-1\\\
\end{array}\right)=\left(\begin{array}[]{cc}\tau_{n}-\tau_{n-1}-1&\tau_{n}\\\
a\tau_{n}&(a+1)\tau_{n}-\tau_{n-1}-1\\\ \end{array}\right).$

So the determinant factors of $N$ are $\Delta_{1}=gcd(\tau_{n},\tau_{n-1}+1)$
and $\Delta_{2}=\tau_{n+1}-\tau_{n-1}-2$. Then the result follows directly
from Theorem 2.3 and Lemmas 5.1–5.2. ∎

Remark 3. Note that $R_{n}(1,0)$ is just the wheel graph $W_{n}$, whose
sandpile group has been determined before, see [9]:

$S(W_{n})=\left\\{\begin{aligned}
&\mathbb{Z}_{\eta_{n}}\oplus\mathbb{Z}_{5\eta_{n}},\ \ \ \mbox{if $n$ is
even;}\\\
&\mathbb{Z}_{\eta_{n+1}+\eta_{n-1}}\oplus\mathbb{Z}_{\eta_{n+1}+\eta_{n-1}},\
\ \ \mbox{if $n$ is odd;}\\\ \end{aligned}\right.$

where
$\eta_{n}=\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^{n}-(\frac{1-\sqrt{5}}{2})^{n})$
are Fibonacci numbers. In [32], the authors considered the sandpile group of
subdivided wheel graphs isomorphic to $R_{n}(2,0)$. It is easy to check that
our results are consistent with the known results in both cases $a=1$ and
$a=2$. Here, we obtain the general result by using a completely different
method. Theorem 5.3 also tells us that the minimum number of generators of
$S(R_{n}(a,0))$ is $2$ if $n\geq 3$.

Now for the sandpile groups of $R_{n}(a,b)$ with $a\geq b>0$. First we
simplify the relation matrix $M$ in Theorem 4.2 as follows: first adding $a+b$
times and $a$ times the first row to the second and the last row,
respectively, we have

$\displaystyle M$
$\displaystyle\sim\left(\begin{array}[]{ccc}a\cdot\frac{\tau_{n}-\tau_{n-1}-1}{a+b}&\tau_{n}&b\cdot\frac{\tau_{n}-\tau_{n-1}-1}{a+b}\\\
a\tau_{n}&\tau_{n+1}-\tau_{n}-1&b\tau_{n}\\\
a\tau_{n}-ab\cdot\frac{\tau_{n}-n}{a+b}&a\cdot\frac{\tau_{n+1}-\tau_{n}-1}{a+b}&ab\cdot\frac{\tau_{n}-n}{a+b}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{ccc}a\rho_{n}&\tau_{n}&b\rho_{n}\\\
a\tau_{n}&(a+b)\rho_{n+1}&b\tau_{n}\\\
a\tau_{n}-ab\tau^{\prime}_{n}&a\rho_{n+1}&ab\tau^{\prime}_{n}\end{array}\right)=M^{\prime},$

where
$\tau^{\prime}_{n}=\frac{\tau_{n}-n}{a+b}=\rho_{n}+\rho_{n-1}+\cdots+\rho_{1}$
are integers.

In order to obtain the determinant factors of $M^{\prime}$, we compute all
order-$2$ minors of $M^{\prime}$. Let $M^{\prime}_{ij;kl}$ denote the minor
corresponding to rows $i,j$ and columns $k,l$. First, by using (3.5) and
$\tau_{n+1}=(a+b+2)\tau_{n}-\tau_{n-1},$ it is easy to check that

$\rho_{n+1}=\tau_{n}+\rho_{n};\ \ \tau_{n}^{2}-(a+b)\rho_{n}\rho_{n+1}=s_{n}.$
$None$

Bearing in mind (5.1) and $\tau_{n}-(a+b)\tau^{\prime}_{n}=n$, we have

$\displaystyle M^{\prime}_{12;12}=-as_{n};\ \ M^{\prime}_{12;13}=0;\ \
M^{\prime}_{12;23}=bs_{n};$ $\displaystyle
M^{\prime}_{13;12}=-a\cdot\frac{as_{n}+nb\tau_{n}}{a+b}=-as_{n}+ab\cdot\frac{s_{n}-n\tau_{n}}{a+b};$
$\displaystyle M^{\prime}_{13;13}=-nab\rho_{n};\ \
M^{\prime}_{13;23}=ab\cdot\frac{s_{n}-n\tau_{n}}{a+b};$ $\displaystyle
M^{\prime}_{23;12}=-nab\rho_{n+1}=-nab(\tau_{n}+\rho_{n});\ \
M^{\prime}_{23;13}=-nab\tau_{n};\ \ M^{\prime}_{23;23}=-nab\rho_{n+1}.$

So it is not difficult to deduce that

$\displaystyle\Delta_{1}(M^{\prime})=gcd(\tau_{n},a\rho_{n},\ a\rho_{n},\
ab\tau^{\prime}_{n})=gcd(\tau_{n},\ \tau_{n-1}+1,\ a\rho_{n},\
ab\tau^{\prime}_{n});$ $\displaystyle\Delta_{2}(M^{\prime})=gcd(nab\tau_{n},\
nab\rho_{n},\ as_{n},\ bs_{n},\ ab\cdot\frac{s_{n}-n\tau_{n}}{a+b});$
$\displaystyle\Delta_{3}(M^{\prime})=nab\cdot s_{n}.$

######  Lemma 5.4.

Suppose that $\tau_{n},\beta_{n},\gamma_{n}$ and $\delta_{n}$ are as above.
Let

$\tau^{\prime}_{n}=\frac{\tau_{n}-n}{a+b},\ \
\beta^{\prime}_{n}=\frac{\beta_{n}-2}{a+b},\ \
\gamma^{\prime}_{n}=\frac{\gamma_{n}-2n+1}{a+b},\textrm{ and
}\delta^{\prime}_{n}=\frac{\delta_{n}-1}{a+b}.$

Then $\tau^{\prime}_{n},\beta^{\prime}_{n},\gamma^{\prime}_{n}$ and
$\delta^{\prime}_{n}$ are integers.

###### Proof.

Note that $\tau_{n},\beta_{n},\gamma_{n}$ and $\delta_{n}$ all satisfy the
recurrence relation $x_{n}=(a+b+2)x_{n-1}-x_{n-2}$. So they can be expressed
as the polynomials of $a+b$. It is easy to show that their constant terms are
$n,2,2n-1$ and $1$, respectively. So the results follow directly. ∎

######  Theorem 5.5.

Let $a\geq b>0$ be integers. Then the sandpile group of $R_{n}(a,b)$ is equal
to

$S(R_{n}(a,b))=\mathbb{Z}_{\Delta_{1}}\oplus\mathbb{Z}_{\frac{\Delta_{2}}{\Delta_{1}}}\oplus\mathbb{Z}_{\frac{\Delta_{3}}{\Delta_{2}}},$

where

$\Delta_{1}=\begin{cases}gcd(gcd(2,a,b)\tau_{m},\
2ab\tau^{\prime}_{m}),&\mbox{if }n=2m\mbox{ is even,}\\\ gcd(\gamma_{m+1},\
ab\gamma^{\prime}_{m+1}),&\mbox{if }n=2m+1\mbox{ is odd};\end{cases}$
$\Delta_{2}=\begin{cases}\tau_{m}\cdot gcd(gcd(a,b)(a+b+4)\tau_{m},ab\cdot
gcd(2m,(a+b+4)\tau^{\prime}_{m}+m)),&\mbox{if }n=2m\hbox{ is even;}\\\
\gamma_{m+1}\cdot gcd(gcd(a,b)\gamma_{m+1},ab\cdot
gcd(2m+1,\gamma^{\prime}_{m+1})),&\mbox{if }n=2m+1\hbox{ is odd};\end{cases}$

and

$\Delta_{3}=\begin{cases}nab(a+b+4)\tau_{m}^{2},&\mbox{if }n=2m\hbox{ is
even;}\\\ nab\gamma_{m+1}^{2},&\mbox{if }n=2m+1\hbox{ is odd}.\end{cases}$

###### Proof.

We only need to show that the determinant factors of $M^{\prime}$ have the
claimed form in the theorem.

(i) If $n=2m$ is even, then by Lemma 5.1, we have

$\displaystyle\tau_{n}=\tau_{m}\beta_{m};\ \
\tau_{n-1}+1=\tau_{m}\beta_{m-1},$ $\displaystyle\rho_{n}=\tau_{m}\gamma_{m};\
\ s_{n}=(a+b+4)\tau_{m}^{2}.$

Recall from Lemma 5.2 that $gcd(\beta_{n},\beta_{n-1})=2$ if $a+b+2$ is even,
and $gcd(\beta_{n},\beta_{n-1})=1$, otherwise. And $\gamma_{n}$ are all odd.
Combining the above facts together, we deduce that

$\displaystyle\Delta_{1}(M^{\prime})$ $\displaystyle=gcd(\tau_{n},\
\tau_{n-1}+1,\ a\rho_{n},\ ab\tau^{\prime}_{n})$
$\displaystyle=\begin{cases}gcd(gcd(2,a)\tau_{m},\
ab\tau^{\prime}_{n}),&\mbox{if }a+b\mbox{ is even};\\\ gcd(\tau_{m},\
ab\tau^{\prime}_{n}),&\mbox{if }a+b\mbox{ is odd}.\end{cases}$
$\displaystyle=gcd(gcd(2,a,b)\tau_{m},\ ab\tau^{\prime}_{n})$
$\displaystyle=gcd(gcd(2,a,b)\tau_{m},\ 2ab\tau^{\prime}_{m}).$

The last equality was obtained by using Lemma 5.3, since

$\tau^{\prime}_{n}=\frac{\tau_{2m}-2m}{a+b}=\frac{\tau_{m}\beta_{m}-2m}{a+b}=\frac{\tau_{m}((a+b)\beta^{\prime}_{m}+2)-2m}{a+b}=\tau_{m}\beta^{\prime}_{m}+2\tau^{\prime}_{m}.$

From the fact that
$gcd(\tau_{n},(a+b)\rho_{n})=gcd(\tau_{n},\tau_{n-1}+1)=\tau_{m}\cdot
gcd(2,a+b)$, we deduce that $gcd(\tau_{n},\rho_{n})=\tau_{m}\cdot
gcd(\beta_{m},\gamma_{m})=\tau_{m}$. So,

$\Delta_{2}(M^{\prime})=\tau_{m}\cdot gcd(nab,\
gcd(a,b)(a+b+4)\tau_{m},\leavevmode\nobreak\
ab\cdot\frac{(a+b+4)\tau_{m}-n\beta_{m}}{a+b}).$

Using Lemma 5.3 again, we have

$\frac{(a+b+4)\tau_{m}-n\beta_{m}}{a+b}=\frac{(a+b+4)((a+b)\tau^{\prime}_{m}+m)-n((a+b)\beta^{\prime}_{m}+2)}{a+b}=(a+b+4)\tau^{\prime}_{m}+m-n\beta^{\prime}_{m}.$

Hence,

$\displaystyle\Delta_{2}(M^{\prime})$ $\displaystyle=\tau_{m}\cdot gcd(\
gcd(a,b)(a+b+4)\tau_{m},\,ab\cdot gcd(2m,(a+b+4)\tau^{\prime}_{m}+m)),$
$\displaystyle\Delta_{3}(M^{\prime})$ $\displaystyle=nab(a+b+4)\tau_{m}^{2}.$

(ii) If $n=2m+1$ is odd, then by Lemma 5.1 again, we have

$\displaystyle\tau_{n}=\gamma_{m+1}\delta_{m},\ \
\tau_{n-1}+1=\gamma_{m+1}\delta_{m-1},$
$\displaystyle\rho_{n}=\gamma_{m+1}\tau_{m},\ \ s_{n}=\gamma_{m+1}^{2}.$

In this case, first note that

$gcd(\tau_{n},(a+b)\rho_{n})=gcd(\tau_{n},\tau_{n-1}+1)=\gamma_{m+1}=gcd(\tau_{n},\rho_{n})$

and

$\tau^{\prime}_{n}=\frac{\gamma_{m+1}\delta_{m}-2m-1}{a+b}=\frac{\gamma_{m+1}((a+b)\delta^{\prime}_{m}+1)-2m-1}{a+b}=\gamma_{m+1}\delta^{\prime}_{m}+\gamma^{\prime}_{m+1}.$

So

$\displaystyle\Delta_{1}(M^{\prime})$ $\displaystyle=gcd(\gamma_{m+1},\
ab\tau^{\prime}_{n})=gcd(\ \gamma_{m+1},\ ab\gamma^{\prime}_{m+1});$
$\displaystyle\Delta_{2}(M^{\prime})$ $\displaystyle=\gamma_{m+1}\cdot
gcd(nab,\ gcd(a,b)\cdot\gamma_{m+1},\
ab\cdot\frac{\gamma_{m+1}-n\delta_{m}}{a+b})$ $\displaystyle=\gamma_{m+1}\cdot
gcd(nab,\
gcd(a,b)\cdot\gamma_{m+1},ab\cdot\frac{(a+b)\gamma^{\prime}_{m+1}+2m+1-n((a+b)\delta^{\prime}_{m}+1)}{a+b})$
$\displaystyle=\gamma_{m+1}\cdot gcd(\ gcd(a,b)\cdot\gamma_{m+1},ab\cdot
gcd(n,\gamma^{\prime}_{m+1}));$ $\displaystyle\Delta_{3}(M^{\prime})$
$\displaystyle=nab\gamma_{m+1}^{2}.$

This completes the proof. ∎

Note that Theorem 5.3 can be viewed as a special case of Theorem 5.5 with
$b=0$. Considering two cases, $a=b$ and $a>b=1$, we obtain the following
corollaries.

######  Corollary 5.6.

Let $a>0$ be an integer. Then for the sandpile group of $R_{n}(a,a)$, we have

$S(R_{n}(a,a))=\mathbb{Z}_{\Delta_{1}}\oplus\mathbb{Z}_{\frac{\Delta_{2}}{\Delta_{1}}}\oplus\mathbb{Z}_{\frac{\Delta_{3}}{\Delta_{2}}},$

where

$\displaystyle\Delta_{1}$ $\displaystyle=\begin{cases}gcd(gcd(2,a)\tau_{m},\
am),&\mbox{if }n=2m;\\\ gcd(\gamma_{m+1},\ a(2m+1)),&\mbox{if
}n=2m+1.\end{cases}$ $\displaystyle\Delta_{2}$
$\displaystyle=\begin{cases}a\tau_{m}\cdot gcd(4m,\
2ma,(a+2)\tau_{m}-2m),&\mbox{if }n=2m;\\\ a\gamma_{m+1}\cdot
gcd(2m+1,\gamma_{m+1}),&\mbox{if }n=2m+1\mbox{ is odd;}\end{cases}$
$\displaystyle\Delta_{3}$
$\displaystyle=\begin{cases}4ma^{2}(a+2)\tau_{m}^{2},&\mbox{if }n=2m;\\\
(2m+1)a^{2}\gamma_{m+1}^{2},&\mbox{if }n=2m+1.\end{cases}$

###### Proof.

(i) Note that if $a=b$, then $\tau_{m}=2a\tau^{\prime}_{m}+m$ and
$\gamma_{m+1}=2a\gamma^{\prime}_{m+1}+2m+1$, that is
$gcd(\gamma_{m+1},a\gamma^{\prime}_{m+1})=gcd(\gamma_{m+1},2m+1).$ So

$gcd(\ gcd(2,a)\tau_{m},\ 2a^{2}\tau^{\prime}_{m})=gcd(\ gcd(2,a)\tau_{m},\
a(\tau_{m}-m))=gcd(\ gcd(2,a)\tau_{m},am)$

and

$gcd(\gamma_{m+1},a^{2}\gamma^{\prime}_{m+1})=gcd(\gamma_{m+1},a(2m+1)).$

Similarly, we can simplify $\Delta_{2}$ and $\Delta_{3}$ to obtain the claimed
expressions. ∎

######  Corollary 5.7.

Let $a>1$ be an integer. Then for the sandpile group of $R_{n}(a,1)$, we have

$S(R_{n}(a,1))=\mathbb{Z}_{\Delta_{1}}\oplus\mathbb{Z}_{\frac{\Delta_{2}}{\Delta_{1}}}\oplus\mathbb{Z}_{\frac{\Delta_{3}}{\Delta_{2}}},$

where

$\displaystyle\Delta_{1}$ $\displaystyle=\begin{cases}gcd(\tau_{m},\
2a\tau^{\prime}_{m}),&\mbox{if }n=2m,\\\ gcd(\gamma_{m+1},\
a\gamma^{\prime}_{m+1}),&\mbox{if }n=2m+1;\end{cases}$
$\displaystyle\Delta_{2}$ $\displaystyle=\begin{cases}\tau_{m}\cdot
gcd(\tau_{m}+4m,\ 2ma),&\mbox{if }n=2m,\\\ \gamma_{m+1}\cdot
gcd(\gamma_{m+1},a\gamma^{\prime}_{m+1}),&\mbox{if }n=2m+1\mbox{ is
odd;}\end{cases}$ $\displaystyle\Delta_{3}$
$\displaystyle=\begin{cases}2ma(a+5)\tau_{m}^{2},&\mbox{if }n=2m;\\\
(2m+1)a\gamma_{m+1}^{2},&\mbox{if }n=2m+1.\end{cases}$

Remark 4. Note that the dual of a polygon ring is just a generalized
suspension of a cycle. So the results obtained above are all applicable to the
generalized suspension of cycles since the sandpile groups of a connected
planar graph and its dual are isomorphic [17].

##  6\. The sandpile group of $T_{n}(a,b)$

Similarly, for the relation matrix $T$ in Theorem 4.2, by first adding $a+b$
times and $a$ times the first row to the second and the last row,
respectively, we have

$\displaystyle T$
$\displaystyle\sim\left(\begin{array}[]{ccc}a\rho_{n}+1&\tau_{n}&b\rho_{n}+1\\\
a\tau_{n}-(a-b)&(a+b)\rho_{n+1}+2&b\tau_{n}+a-b\\\
a\tau_{n}-ab\tau^{\prime}_{n}-a\tau_{n-1}+a&a\rho_{n+1}-\tau_{n}+\tau_{n-1}&ab\tau^{\prime}_{n}-b\tau_{n-1}+a\end{array}\right)$
$\displaystyle\sim\left(\begin{array}[]{ccc}a\rho_{n}+1&\tau_{n}&b\rho_{n}+1\\\
a\tau_{n}-(a-b)&(a+b)\rho_{n+1}+2&b\tau_{n}+a-b\\\ a-b-
ab\tau^{\prime}_{n}&-b\rho_{n+1}-1&ab\tau^{\prime}_{n}-b\tau_{n}\end{array}\right)$
$\displaystyle\sim\left(\begin{array}[]{ccc}a\rho_{n}+1&\tau_{n}&b\rho_{n}+1\\\
a\tau_{n}-(a-b)&(a+b)\rho_{n+1}+2&b\tau_{n}+a-b\\\
a\tau_{n}-ab\tau^{\prime}_{n}&a\rho_{n+1}+1&ab\tau^{\prime}_{n}+a-b\end{array}\right)=T^{\prime}.$

It is easy to see that

$\Delta_{1}(T^{\prime})=gcd(\tau_{n},\ a\rho_{n}+1,\ ab\tau^{\prime}_{n},\
a-b);\ \
\Delta_{3}(T^{\prime})=nabs_{n}+\frac{4nab}{a+b}+\frac{(a-b)^{2}}{a+b}\tau_{n}.$

But the expression for $\Delta_{2}(T^{\prime})$ is cumbersome. So we only give
the result for $a=b$. In this case

$\displaystyle T^{\prime}_{12;12}=-T^{\prime}_{12;23}=as_{n}+2;\ \
T^{\prime}_{12;13}=0;$ $\displaystyle
T^{\prime}_{13;12}=a\cdot\frac{s_{n}-n\tau_{n}}{2}+1;\ \
T^{\prime}_{13;13}=-na(a\rho_{n}+1);$ $\displaystyle
T^{\prime}_{13;23}=a\cdot\frac{s_{n}-n\tau_{n}}{2}-as_{n}-1;$ $\displaystyle
T^{\prime}_{23;12}=T^{\prime}_{23;23}=-na(a\rho_{n+1}+1);\ \
T^{\prime}_{23;13}=-na^{2}\tau_{n}.$

Hence

$\displaystyle\Delta_{1}$ $\displaystyle=gcd(\tau_{n},\ a\rho_{n}+1,\
a^{2}\tau^{\prime}_{n});$ $\displaystyle\Delta_{2}$
$\displaystyle=gcd(na^{2}\tau_{n},\ as_{n}+2,\
na(a\rho_{n}+1),\frac{as_{n}+2-na\tau_{n}}{2});$ $\displaystyle\Delta_{3}$
$\displaystyle=na(as_{n}+2).$

######  Theorem 6.1.

Let $a>0$ be an integer. Then for the sandpile group of $T_{n}(a,a)$, we have

$S(T_{n}(a,a))=\mathbb{Z}_{\Delta_{1}}\oplus\mathbb{Z}_{\frac{\Delta_{2}}{\Delta_{1}}}\oplus\mathbb{Z}_{\frac{\Delta_{3}}{\Delta_{2}}},$

where

$\Delta_{1}=\left\\{\begin{aligned} &gcd(\ \beta_{m}/2,\ am);\ \ \mbox{if
$n=2m$;}\\\ &gcd(\ \delta_{m},\ a(2m+1)),\ \ \mbox{if
$n=2m+1$}.\end{aligned}\right.$ $\Delta_{2}=\left\\{\begin{aligned}
&\beta_{m}/2\cdot gcd(na,\ \beta_{m},\beta_{m}/2-na\tau_{m});\ \ \mbox{if
$n=2m$;}\\\ &\delta_{m}\cdot
gcd(na,\frac{\beta_{m}+\beta_{m+1}}{2},\frac{\beta_{m}+\beta_{m+1}-2na\gamma_{m+1}}{4}),\
\ \mbox{if $n=2m+1$ is odd;}\end{aligned}\right.$

and

$\Delta_{3}=\left\\{\begin{aligned} &ma\beta_{m}^{2};\ \ \mbox{if $n=2m$;}\\\
&(2m+1)a\cdot\frac{\delta_{m}(\beta_{m}+\beta_{m+1})}{2},\ \ \mbox{if
$n=2m+1$.}\end{aligned}\right.$

Remark 5. Note that the polygon ring is obtained by identifying two free edges
of a polygon chain. It is known that any free edge can generate the sandpile
group of the polygon chain. So we wonder whether the result we obtained above
may hold in a more general setting. That is, let $H$ be a plane graph with
cyclic sandpile group, and $e$ and $f$ be two generating edges on the outer
face. Let $G$ be the graph obtained from $H$ by identifying $e$ and $f$. Does
it follow that the sandpile group of $G$ is the direct sum of at most three
cyclic groups?

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