On the sandpile group of the cone of a graph

Abstract

The majority of graphs whose sandpile groups are known are either regular or simple. We give an explicit formula for a family of non-regular multi-graphs called thick cycles. A thick cycle graph is a cycle where multi-edges are permitted. Its sandpile group is the direct sum of cyclic groups of orders given by quotients of greatest common divisors of minors of its Laplacian matrix. We show these greatest common divisors can be expressed in terms of monomials in the graph's edge multiplicities.

Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields

“…(1) Since the columns of the Laplacian ∆ add up to 0, we have that f v (1) = 0; moreover, since both DB(n, d) and Ktz(n, d) are Eulerian, in both cases the rows of ∆ also add up to 0, so that v∈Zn f v (x) = 0 (these claims are also easily verified directly). Part (2) is obvious from the observation that e v+1 − e v = (x − 1)x v , and to see (3), simply note that from (1), after multiplication by x − 1, we obtain g 0 (x) + · · · + g n−1 = 0.…”

“…For more details and background, see, for example, [21,29,1] for the undirected case and [15,33] for the directed case.…”

“…For some examples, see [8,16,26,27,9,10,30,22,5], and the references in [1]. Here, we determine the critical group of the generalized de Bruijn graphs DB(n, d) and generalized Kautz graphs Ktz(n, d) (which are in fact both directed graphs), thereby extending and completing the results from [19] for the binary de Bruijn graphs DB(2 ℓ , 2) and Kautz graphs Ktz((p − 1)p ℓ−1 , p) for primes p, and [4] for the classical de Bruijn graphs DB(d ℓ , d) and Kautz graphs Ktz((d − 1)d ℓ−1 , d).…”

Graphs with few trivial critical ideals

“…The critical group is especially interesting for connected graphs, since its order is equal to the number of spanning trees of the graph. The critical group has been studied intensively over the last 30 years on several contexts: the group of components [25,26], the Picard group [11,12], the Jacobian group [11,12], the sandpile group [5,17], chip-firing game [12,27], or Laplacian unimodular equivalence [22,28]. The book of Klivans [24] is an excellent reference on the theory of sandpiles and its connections to other combinatorial objects like hyperplane arrangements, parking functions, dominoes, etc.…”

On the Sandpile Model of Modified Wheels I

“…The sandpile model was first defined in a book for school teachers in 1975 by Engel [1], who called it the chip firing game. It was rediscovered by Bak-TangWiesenfeld as an example of self organized criticality.…”