On the sandpile model of modified wheels II

Abstract We investigate the abelian sandpile group on modified wheels W ˆ n {\hat{W}}_{n} by using a variant of the dollar game as described in [N. L. Biggs, Chip-Firing and the critical group of a graph, J. Algebr. Comb. 9 (1999), 25–45]. The complete structure of the sandpile group on a class of graphs is given in this paper. In particular, it is shown that the sandpile group on W ˆ n {\hat{W}}_{n} is a direct product of two cyclic subgroups generated by some special configurations. More precisely, the sandpile group on W ˆ n {\hat{W}}_{n} is the direct product of two cyclic subgroups of order a n {a}_{n} and 3 a n 3{a}_{n} for n even and of order a n {a}_{n} and 2 a n 2{a}_{n} for n odd, respectively.


Introduction
The abelian sandpile model (ASM) was defined in [2] for any graph. This model is a prime example of selforganized criticality [3], which has transformed the understanding of how complexity arises in nature. The ASM is related to the chip-firing game introduced by Spencer [4].
The algebraic structure of the sandpile group has been completely determined for the following family of graphs: wheel graphs, trees, regular trees, thick trees, wired regular trees, cycles, complete graphs, multipartite complete graphs, wheels, some thick wheels, the square of the cycle ( ) C n 2 , dihedral group graph, Möbius ladder, n 3 twisted bracelets, generic threshold graphs, the Cartesian products ( ) × ( ) P C n , and ( ) × ( ) K C m n . The abstract structure of the sandpile group has been partially described for the hypercube and the Cartesian product of complete graphs.
However, the combinatorial structure of the recurrent configurations has only been described for a thick tree (a graph is a thick tree if it is chordal and every two separators are either equal or disjoint). In [1], it has been shown that the set of critical configurations can be used to give the structure of an abelian group by giving critical configurations for a wheel graph W n . For general references on the ASM (denoted by SP (G,q)), see, e.g., [5][6][7][8][9][10][11][12][13][14] and on the critical group (denoted by K(G)), see for instance [15][16][17][18][19][20][21][22][23][24]. The definitions and statements of theorems presented in this section have been taken from [1]. We only consider undirected graphs.
The sandpile model is a discrete model (cellular automaton) originally defined on a rectangular grid of the standard square grid in the plan which was extended on the graphs as well.
An (undirected) ASM begins with any connected undirected graph G with vertex set ( ) V G and edge set ( ) E G , together with a distinguished vertex ∈ ( ) q V G called the central or hub vertex. To each vertex of the graph G, we associate a value (grains of sand, chips) referred to as the (initial) configuration of the sandpile. The dynamics of the model at iteration ∈ i are then defined as follows: -Choose a random vertex according to some probability distributions (usually uniform).
-Add one grain of sand to this vertex while letting the grain numbers for all other vertices unchanged.
-If all vertices are stable. In this case, continue with the next iteration.
-If at least one vertex is unstable, then whole configuration is said to be unstable. In this case, choose any unstable vertex at random. Topple this vertex by reducing its grain number by number of the neighbors and by increasing the grain numbers of each of its direct neighbors by one. -Due to the redistribution of grains, the toppling of one vertex can render other vertices unstable. Thus, repeat the toppling procedure until all vertices of eventually become stable and continue with the next iteration.
A dollar game on a graph G starts with dollars at each vertex of a graph. At each step of the game a vertex is "fired," that is, ( ) v deg dollars move from the fired vertex to the adjacent vertices, one dollar going along each edge incident with the fired vertex. A vertex v can be fired if and only if the number of dollars currently held at v is at least Let s be a configuration of the game on the graph G. By a configuration we mean that s is a function defined on the vertices ( ) V G such that ( ) s v is the number of dollars at each vertex v. Suppose that S is a non-empty finite sequence of (not necessarily distinct) vertices of G, such that, starting from s, the vertices can be fired in the order of S. If v occurs ( ) x v times in S, x shall be referred to as the representative vector for s. After "firing" a change in the configuration occurs. The configuration ′ s obtained after the sequence of firings S is given by , is the number of edges joining v and w. Each time a vertex v is fired it loses dollars. It should be noted that ′ s always exists and is unique.
The Laplacian matrix L is given by and is adjacent to , The relationship between s and ′ s can be written concisely if we define the Laplacian matrix Q as follows: In terms of Q the relationship between s and ′ s is given as Definition 1.1. The family of graph denoted by Ŵ n is constructed from an n-wheel W n by adding one vertex on each of the cycle edge of the wheel. In particular, let W n be the wheel with vertices … q v v v , , , , n  Definition 1.2. A configuration s is said to be stable if it satisfies the following inequality: Note that a configuration with 2 dollars at each vertex is stable for W n , see [1], but not for Ŵ n .
A sequence of firing is q-legal if and only if each occurrence of a vertex ( ≠ ) v q follows a configuration t with ( ) ≥ ( ) t v v deg and each occurrence of q follows a stable configuration. A configuration r of the dollar game on a graph is said to be recurrent if there is a q-legal sequence for r which leads to the same configuration.
A configuration which is both recurrent and stable is referred to as the critical configuration. Note that not all stable configurations are critical (e.g., the configuration with zero dollars at every vertex is stable but not recurrent).
The theory of dollar game is based on a confluence property, i.e., if we start with a given configuration s, then there may be many different sequences which are possible starting from s, but all of these sequences will lead to the same outcome. For the proof of this property, we direct the reader to [1].
We shall study a variant of the dollar game in which the fixed central vertex q is allowed to go into debt. The central vertex q is fired if and only if no other firing is possible. Thus, in this variant, a configuration s is an integer-valued function satisfying: Equivalently, any dollars going into the center are ignored. We shall refer to a sequence as a proper sequence if it does not contain the central vertex q. Given a configuration s and ∈ ( ) { } v V G q \ , toppling unstable non-central vertices on a finite connected graph until no unstable non-central vertex remains leads to a unique stable configuration ′ s , which is called the stabilization of s. Given two stable configurations s and w, we can define the operation → ( + )′ s w z w ⁎ , corresponding to the vertex-wise addition of dollars followed by the stabilization of the resulting sandpile. Under the operation ⁎, the set of critical configurations forms an abelian group isomorphic to the cokernel of the reduced graph Laplacian ′ Q , i.e., ( to which the former is isomorphic, is most commonly referred to as the sandpile group. Other common names for the same group are critical group, Jacobian group, or (less often) Picard group. Note, however, that some authors only denote the group formed by the recurrent configurations as the sandpile group, while reserving the name Jacobian group or critical group for the (isomorphic) group defined by ( Given the isomorphisms stated above, the order of the sandpile group is the determinant of ′ Q , which by the matrix tree theorem is the number of spanning trees of the graph. Let > l 1 be any positive integer. Then ⋅ l a denotes the ⁎-sum of l copies of a configuration a in ( ) SP G q , . Equivalently, ⋅ l a is the unique critical configuration ′ = ( + + ⋯+ ) s γ a a a , where the + sign represents vector addition.
For example, let = ( ) a 2, 0, 2, 1, 2, 1, 2, 1 be a configuration on Ŵ 4 and = l 3, then ⋅ = + + a a a a 3 is equal to the following critical configuration ⋅ = ( ) a 3 1, 0, 1, 1, 2, 1, 2, 1 . In this section, we give our main theorem by providing the complete structure of the sandpile group on a class of graphs. In particular, we show that the sandpile group is a direct product of two cyclic groups generated by some special configurations. That is, if n is even, these two subgroups are of order a n and a 3 n , and if n is odd they are of order a n and a 2 n , respectively.
The vertices of the graph Ŵ n are numbered as … − n 0, 1, 2, 3, , 2 1, where the vertex numbered 0 has degree 3. Let a n be a sequence of positive integers with initial conditions = = = = a a a a 0, 1, Case I: If = n k 2 is an even number and define the following four configurations on the graph Ŵ n as:  It can be easily verified that the following sequence is aq legal sequence for this configuration and results in ′ c n n n n n n 1, 2, 3, ,2 1, 0, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, ,2 1, 0, 1, 2, 3, ,2 1, 1, 2, 3, 4, ,2 1, 0, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, ,2 1, 0, 1, 2, ,2 1. It can be easily verified that the following sequence is aq legal sequence for this configuration and results in c Proof. Clearly ′ c is stable and the following sequence of vertices isq legal for ′ c … − q n , 2, 3, 4, , 2 1, 0, 1, since each vertex is fired once, ′ c is a recurrent configuration and hence critical. The following sequence of firings is aq legal sequence and leads to identity for ⋅ ′ c 3 . By definition c is stable and the given sequence of vertices isq legal for c … − n 0, 1, 2, 3, , 2 1, which shows that c is a recurrent configuration and hence critical. Similarly, it can be checked that the following sequence is aq legal sequence and leads to identity for ⋅ c 3 . Proof. Let us consider a configuration t that satisfies the aforementioned conditions and let an odd numbered vertex say v i has zero dollars where index i is odd. In the very first round of firings, the vertex v i does not fire but receives 2 dollars from its neighboring vertices (as they were destabilized initially). In other rounds of firings, the number of dollars at odd numbered vertices remains constant because the odd numbered vertices are not directly connected to the central vertex so whatever they loose they gain it back through their neighbors. □ Since all the even numbered vertices had equal dollars initially, they stabilize at the same time. Once this happens, start firing the odd numbered vertices. The vertex v i then loses each of its dollar to its neighbors − v i 1 and + v i 1 which are even numbered vertices. If the neighbors of v i are destabilized after receiving a dollar each from v i , then fire all the destabilized even numbered vertices. As a result v i gains 2 dollars and is destabilized. Repeat the procedure of firing odd and even numbered vertices alternately until firing of v i does not destabilize its neighbors. 2, 1, 1, 1, 2, 0, 2, 1 , 16 1, 1, 2, 1, 1, 1, 2, 1 , 18 2, 1, 1, 1, 1, 1, 2, 1 .
On the sandpile model of modified wheels II  1535 Theorem 2.7. If n is even, the abelian sandpile group ( ) SP W q , n is isomorphic to the direct product of two cyclic groups of orders a n and a 3 n . In particular,  Every multiple of a is symmetrical with respect to − − π n 1, 1 , that is, ( ⋅ )( ) = ( ⋅ )( The required configuration cannot be a configuration in which every rim vertex has the same number of dollars because such a configuration is not critical for this particular family of graphs. Therefore, after deleting the non-critical configurations we are left with the configurations ( … ) 2, 0, 2, 0, , 2, 0 and (2, 1, … ) 2, 1, , 2, 1 . By using Lemma 2.3, the choice ( … ) 2, 1, 2, 1, , 2, 1 is ruled out and hence ( … ) 2, 0, 2, 0, , 2, 0 is the required critical configuration, which is also the zero element of the sandpile group ( ) SP W q , n . It follows that the subgroup generated by a and b is the direct sum of the cyclic groups, which completes the proof.
Case II: If = + n k 2 1 is an odd number and define some configurations on Ŵ n .  .
It can be verified that the following sequence is a q-legal sequence for this configuration and results in f. Proof. By definition, f is a stable configuration and since the following sequence of vertices … 0, 1, 2, 3, , − n 2 1 is q-legal for f, therefore, f is a recurrent configuration and hence critical. The following sequence is a q-legal sequence for ⋅ = + such that every even numbered vertex has the same number of dollars, the number of dollars at v 1 and − v n 2 1 is the same, the number of dollars at remaining odd vertices is zero, multiplying by α destabilizes all the vertices, then after stabilizing v n will always have zero dollars.
Proof. It is easy to identify that the only configurations that satisfy the aforementioned conditions are d and e given as: 0 mod 2, 1, 1, 2 1, 0, otherwise, 2, 0 mod 2, 0, otherwise.
The argument given below is based on the fact that if we start with a given configuration p, then there may be many different sequences of firings which are possible from p, but they all lead to the same outcome. So we define certain concrete configurations that can be reached starting from ⋅ α d and ⋅ α e which will lead to the desired result. First, we shall give the proof for configuration d.
In the initial round of firings, all vertices except the vertices with zero dollars fire and loose a dollar each. While the vertices with zero dollars gain 2 dollars each. Fire the vertices in the sequence … − n 1, 2, , 2 1 a number of times till the following configuration is obtained: