On the Sandpile Group of Circulant Graphs

Transcription

1 On the Sandpile Group of Circulant Graphs Anna Comito, Jennifer Garcia, Justin Rivera, Natalie Hobson, and Luis David Garcia Puente (Dated: October 9, 2016) Circulant graphs are of interest in many areas of mathematics, particularly geometric group theory, because of the beautiful cyclic symmetries they display. These graphs, denoted C n (A), can be defined by a set of integer generators, A = {a 1,..., a m }, on a fixed number of n vertices. The adjacency matrices of circulant graphs are also circulant. It has been shown that the number of spanning trees of circulant graphs with a given set of generators follows a recursive formula. One may ask if this structure is reflected in the corresponding sandpile group as well. Previous results on the sandpile group for C n (1, 2) seem to infer this may be the case. In this project, we create a database of isomorphism classes and sandpile groups for a large collection of circulant graphs. From our database, we conclude that such nice structure is not always preserved in the sandpile group of circulant graphs and hence the sandpile group is a more elusive graph invariant. We further focus our attention to the case C n (1, 3) in order to determine the explicit structure of the sandpile group for this family of graphs. I. INTRODUCTION A. The Sandpile Group The Abelian Sandpile Model is a model which displays self-organized criticality. It was first introduced by Bak, Tang and Wiesenfeld in 1987 as the simplest model to display such behavior [1]. The model was originally defined on a finite grid, but can be extended more generally to finite graphs as well. In the model, grains of sand are placed on the vertices of a finite graph. A vertex is said to be stable if the number of grains of sand at the vertex is strictly less than the degree of the vertex. If a vertex becomes unstable - that is, if the number of grains of sand is greater than or equal to the degree - the vertex will topple, sending one grain of sand along each of its adjacent edges. To ensure a sandpile stabilizes in a finite number of steps, we distinguish a vertex s to be the sink. The sink may collect

2 2 any number of grains of sand and is never considered unstable. A stable configuration is recurrent if it is accessible from any other sandpile via a sequence of sand additions and topplings. Two sandpiles on a graph Γ may be added vertex-wise, and then stabilized under the binary operation stable addition, denoted. The collection of recurrent sandpiles under stable addition form a group called the sandpile group. The sandpile group of the graph Γ is denoted S(Γ). The sandpile group can be determined by the Laplacian matrix of Γ. Definition I.1. For a graph Γ with n vertices {v 1,...v n }, the Laplacian matrix, L(Γ), is an n n matrix defined as L(Γ) = D A, where D is the diagonal matrix whose (i, i) entry is the degree of v i and A is the adjacency matrix of Γ. The reduced Laplacian, L(Γ), is obtained by deleting the row and column of L(Γ) associated with the sink. Theorem I.1. The Fundamental Theorem of Sandpile Groups [3] tells us that S(Γ) = Z n / L(Γ)Z n. Furthermore, given a graph with reduced Laplacian L(Γ), if we let diag(k 1, k 2,..., k n ) be its Smith normal form, where the diagonal entries are called invariant factors, then S(Γ) = Z k1 Z k2 Z kn. Theorem I.2 (Kirchhoff's Matrix Tree Theorem [2]). Let Γ be an undirected connected graph and L be its Laplacian matrix. Then, κ(γ) = Det( L(Γ)) = S(Γ), where κ(γ) is the number of spanning trees on Γ. B. Circulant Graphs Let n be a positive integer and {a 1,..., a m } be a set of m integers such that 1 a j n for each j. We define the circulant graph C n (a 1,..., a m ) to be the graph with vertex set {v 0, v 1,..., v n 1 } and edges connecting vertex v i to each vertex v i+aj, where addition on the indices is taken modulo n. We will assume that gcd(a 1,..., a m, n) = 1 in order to ensure that the resulting graph is connected. Figure 1 shows several examples of circulant graphs.

3 3 FIG. 1: The Circulant Graphs C 7 (1, 3), C 8 (2, 3), and C 13 (2, 5) A natural way of studying circulant graphs is through their adjacency matrices, which are also circulant. An n n matrix C is a circulant matrix if it has the following form: c 0 c 1 c 2 c n 1 c n 1 c 0 c 1 c n 2 C = c n 2 c n 1 c 0 c n c 1 c 2 c n 1 c 0 Note that a circulant matrix is completely determined by its first row, since every other row can be obtained by shifting the entries in the first row to the right. Note also that a circulant graph Γ must be a regular graph. Indeed, each vertex has the same degree determined by the number of generators. Thus, the Laplacian matrix of Γ must also be a circulant matrix. Circulant graphs are a large family of graphs. In some cases of n and {a 1,..., a m } these graphs have been studied under the names of generalized crown graphs, complete bipartide graphs and Möbius latter graphs. Such a large family of graphs makes it a difficult class to study, yet the structure of circulant graphs gives it undeniable beauty. As a direct consequence, it is a difficult yet intriguing task to compute S(C n (a, b)).

4 4 II. PREVIOUS WORK AND MOTIVATION It can be shown that crown graphs and Möbius latter graphs are circulant graphs. For these two cases, the sandpile group is known. In the language of circulant graphs, the only previous work in studying S(C n (a 1,..., a m )) was by Hou, Woo, and Chen [4], who focused on S(C n (1, 2)). They determined that the sandpile group of this particular graph had the following form: S(C n (1, 2)) = Z gcd(n,fn) Z Fn Z nfn, gcd(n,fn) where F n is the Fibonacci sequence defined recursively by F n = F n 1 + F n 2 with roots F 0 = 1 and F 1 = 1. Additionally, Zhang, Yong, and Golin [5] give an explicit formula for the number of spanning trees on a circulant graph. S(C n (a 1,..., a m )) = nx n 2, where x n is defined recursively. These examples demonstrate that the elegant geometric structure of the number of spanning trees carries over into the sandpile group. With these findings, one may ask whether this well-behaved structure holds true for the sandpile group of circulant graphs in general. As we will show in Section IV, the structure of the number of spanning trees on C n (1, 3) does not decompose nicely in S(C n (1, 3)) as it does in S(C n (1, 2)). In fact, the structure is far less predictable than in this previously computed case. III. CLASSES OF ISOMORPHIC CIRCULANT GRAPHS By the Fundamental Theorem of Sandpile Groups, we know that if two graphs are isomorphic, then their sandpile groups are isomorphic. This allows us to investigate the sandpile groups of a smaller collection of circulant graphs, rather than looking at the family as a whole. Explicitly, we can classify generators {a 1,..., a m } in order to determine isomorphism classes of circulant graphs. For example, as a first step in classifying the generators a and b for a fixed n to determine the isomorphism classes of graphs C n (a, b), we have the following isomorphisms: C n (a, b) = C n (a 1, b) = C n (a, b 1 ) = C n (a 1, b 1 ),

5 5 where a 1 and b 1 are the respective inverses in Z n under addition modulo n. See Figure 2 for an example, where we list these isomorphisms in a particular circulant graph with n = 6 vertices. Additionally, this tells us that given a fixed n and number of generators m, we only need to consider generators whose values are less than or equal to n/2. That is, in order to study C n (a 1,..., a m ), we need only focus on graphs with generators 1 a j n/2. FIG. 2: C 6 (1, 2) = C 6 (1, 4) = C 6 (2, 5) = C 6 (4, 5) In addition to these isomorphisms which can be directly seen in the relationships between the generators and their inverses, other isomorphisms can be characterized by the following remark. Remark: Two graphs C n (a, b) and C n (c, d) are isomorphic if we can find a Z n automorphism ϕ such that ϕ(a) c and ϕ(b) d or ϕ(a) d and ϕ(b) c. Example III.1. Consider the two graphs C 7 (1, 2) and C 7 (2, 3). Consider the isomorphism ϕ : Z 7 Z 7 defined by ϕ(v) = 2v. The image of the generator 1 Z 7 is ϕ(1) = 2. Also, the image of the generator 2 Z 7 is ϕ(2) = 4. From the above remarks, since 4 1 = 3 Z 7, we can conclude that the image of the gererator 2 Z 7 is equal to 3 Z 7. We see that ϕ induces the isomorphism: C 7 (1, 2) = C 7 (2, 3).

6 6 By classifying isomorphic circulant graphs as described, we can focus on one graph in each isomorphism class for any given n. By studying the table in Figure 3, we notice that all circulant graphs with five vertices are isomorphic. Thus, there is only one sandpile group we must compute in this case. For n = 10 there are five non-isomorphic circulant graphs to be studied. FIG. 3: The number of non-isomorphic circulant graphs for a given n. IV. THE DATABASE OF CIRCULANT GRAPHS It is useful to catalogue isomorphism classes of circulant graphs with a fixed number of vertices and list certain properties for each class. In order to catalogue this information, we created a code in Sage that exports this information into a text file. We use MySQL to store this data. The database contains twenty-one tables labeled n6 through n27. Within each of these tables, there are two subcatagories labeled graphclass and invfactors. The category graphclass shows the circulant graphs which are isomorphic to one another for any given n. The other category, invfactors, prints the number of non-isomorphic graphs, the invariant factors, the prime decomposition of the invariant factors, and the order of the sandpile group for each isomorphism class in graphclass. It also presents the instances when the circulant

7 7 graphs are not isomorphic but the sandpile groups of the circulant graphs are isomorphic. For example, for n = 8 the database produces the following tables: FIG. 4: Menu showing possible values for n FIG. 5: Menu upon choosing n8

8 8 FIG. 6: Menu upon choosing graph class FIG. 7: Menu upon choosing inv factors

9 9 Our code finds all combinations of generators that are less than or equal n and displays 2 the list of all other isomorphic graphs. If a graph is not isomorphic to any other graph, the word none will appear. The code displays the number of non-isomorphic graphs for each value of n. If two sandpile groups are isomorphic, the code will check if the graphs are isomorphic and then display the result. A representative graph of the isomorphic graph class is chosen and the code computes the invariant factors along with the prime decompisition of the invariant factors in an effort to find any patterns in their structure from graph to graph. It also computes the order of the sandpile group. A. Observations From the Database By the Fundamental Theorem of Sandpile Groups, we know that if two graphs are isomorphic, then their sandpile groups are also isomorphic. Our database shows that the converse is true for small values of n. Theorem IV.1. For n 19, C n (k 1, k 2,..., k i ) = Cn (j 1, j 2,..., j i ) if and only if S(C n (k 1, k 2,..., k i )) = S(C n (j 1, j 2,..., j i )). This, however, does not hold for larger values of n, as the following example shows. Example IV.1. For n = 20, we notice that C 20 (1, 2, 4, 9, 10) C 20 (1, 6, 8, 9, 10), but S(C 20 (1, 2, 4, 9, 10)) = S(C 20 (1, 6, 8, 9, 10)) = Z 19 Z 3 95 Z Z This is a particularly interesting example because it demonstrates that the relationship between the structure of the graphs is not always preserved in the graphs associated sandpile groups.

10 10 V. THE SANDPILE GROUP OF C n (1, 3) As described in Section II, Hou, Woo, and Chen made some elegant discoveries about the sandpile group of C n (1, 2) in [4]. In an effort to extend their results for the more general case of S(C n (a, b)), we first turn our attention to the case of S(C n (1, 3)). A. A System of Relations for the Generators of S(C n (1, 3)) We begin by finding a bound on the number of invariant factors for S(C n (1, 3)). Using the information from the database, we noticed behavior in the number invariant factors of S(C n (1, 3)). In Theorem V.1 we prove that the number of invariant factors of S(C n (1, 3)) is less than or equal to five. We generalize this result in Section VI, where we compute an upper bound for the number of invariant factors of S(C n (a, b)). Theorem V.1. For S(C n (1, 3)) the maximum number of invariant factors is 5. Proof. Consider the family of circulant graphs S(C n (1, 3)). By the Fundamental Theorem of Sandpile Groups, we can describe the sandpile group on C n (1, 3) in the following manner: where i is given by, Z S(Γ) = Z n /span( 1,..., n ), i = d i x i v j adjacent to v i x j, where d i the degree of v i and x i = (0,..., 0, 1, 0,..., 0) Z n, with 1 in the i th position. Note that i is the i th row of the Laplacian matrix, L(C n (1, 3)). In our case, the following relation is true for n 7: i = 4x i x i 1 x i 3 x i n+1 x i n+3. Now let x i be the image of x i in Z n /span( 1,..., n ). Then we can solve for x i to get: x i = 4x i 3 x i 2 x i 4 x i 6. This relationship between the x i s can also be seen by observing the adjacent vertices to vertex v i 3 in C n (1, 3). See Figure 8 for an illustration.

11 11 From this equation, it can be seen that for i 7, x i can be expressed as a linear combination of the values in {x 1,..., x j }, where j < 7. Note that {x 1,..., x j } = 6. Now, letting v 6 be the sink of our graph, then x 6 becomes zero in S(C n (1, 3)). Thus, for S(C n (1, 3)), there are at most 6 1 = 5 generators. FIG. 8: Visual representation of the vertices associated with x i Example V.1. The above is most readily understood with an example. Consider C 7 (1, 3).

12 12 Using the above equation for x i, we get: x 1 = 4x 6 x 7 x 5 x 3 x 2 = 4x 7 x 8 x 6 x 4 x 3 = 4x 8 x 1 x 7 x 5 x 4 = 4x 1 x 8 x 2 x 6 x 5 = 4x 2 x 1 x 3 x 7 x 6 = 4x 3 x 2 x 4 x 8 x 7 = 4x 4 x 5 x 3 x 1 x 8 = 4x 5 x 4 x 6 x 2. As before, we designate v 6 as the sink. Note that x 7 and x 8 are simply linear combinations of {x 1,..., x 6 }. Since x 6 represents the sink, we have x 6 = 0. So, we are left with the following five generators of S(C n (1, 3)): x 1 = 4x 6 x 7 x 5 x 3 x 2 = 4x 7 x 8 x 6 x 4 x 3 = 4x 8 x 1 x 7 x 5 x 4 = 4x 1 x 8 x 2 x 6 x 5 = 4x 2 x 1 x 3 x 7. B. Further Characterization of the Invariant Factors of S(C n (1, 3)) In the previous section, Theorem V.1 provided an upper bound on the number of invariant factors of S(C n (1, 3)). Further investigation allows us to characterize those n for which the sandpile group of C n (1, 3) has 2, 3, 4, or 5 invariant factors. By examining S(C n (1, 3)) for n = 6 through n = 200, we noticed that, if n is a multiple of 8 or 30, then S(C n (1, 3)) will have 5 invariant factors. If n is a multiple of 10 or 25,

13 13 S(C n (1, 3)) will have 4 invariant factors. If n = 10k 5j for some k, j N, then S(C n (1, 3)) will have 3 invariant factors. Lastly, if n is prime or n 2mod4, then S(C n (1, 3)) will have 2 invariant factors. It is important to note that these relationships do not classify all possible values of n. There are many n N which do not fit into the relations described above and for which S(C n (1, 3)) has 2, 3, 4, or 5 invariant factors. Furthermore, if a given n value can be described by more than one of the above classifications, the classification which gives a greater number of invariant factors will be given preference. An example follows. Example V.2. Consider n = 80. In this case, n is a multiple of 8 and a multiple of 10. S(C 80 (1, 3)) has the larger number of invariant factors determined by the above classifications. Indeed, S(C 80 (1, 3)) has 5 invariant factors. VI. THE SANDPILE GROUP OF C n (a, b) We generalize Theorem V.1 from Section V and determine a bound for the number of invariant factors for S(C n (a, b)). Theorem VI.1. For S(C n (a, b)), the maximum number of invariant factors is 2b 1. Proof. Using the same setup as in the proof of Theorem V.1, we obtain the following relationship for the generators x i of Z n /span( 1,..., n ). x i = 4x i b x i b+a x i b a x i 2b. We can visualize this relationship by observing the adjacent vertices of vertex v i b. See Figure 9. Again, note that determining x i requires that we define x i 1, x i 2,..., x i 2b. Let x i 2b represent the sink vertex of S(C n (a, b)). Thus, for S(C n (a, b)), we need at most {x i 1, x i 2,..., x i 2b 1 } = 2b 1 generators.

14 14 FIG. 9: A visual representation of the vertices associated with x i in C n (a, b) VII. CONCLUSION There is still little known about the sandpile group of the family of circulant graphs, but the results which have been proven are enticing. In an effort to further understand this family of graphs and their sandpile group, we compiled a collection of them for n 27 into a database which contains relevant information on their group structure, such as their invariant factors and orders, and identifies isomorphisms between graphs. We are also able to make several observations about S(C n (1, 3)) from this database. We expect that, along with our database and the observations we have made thus far, we will soon be able to find an explicit forumla for S(C n (1, 3)) and S(C n (a, b)).

15 15 References [1] Bak, P., Tang, C., and Wiesenfeld, K. Self-organized criticality: An explanation of the 1/ f noise. Phys. Rev. Lett. 59 (July 1987), [2] Biggs, N. Algebraic Graph Theory, vol. 64. Cambridge University Press, [3] Dhar, D. Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64 (Apr 1990), [4] Hou, Y., Woo, C., and Chen, P. On the sandpile group of the square cycle c n [5] Zhang, Yong, and Golin. Chebyshev polynomials and spanning tree formulas for circulant and related graphs. Discrete Mathematics 298 (August 2005),