Journal of Combinatorial Theory, Series A

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1 Journal of Combinatorial Theory, Series A Contents lists available at ScienceDirect Journal of Combinatorial Theory, Series A Sandpile groups and spanning trees of directed line graphs Lionel Levine 1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States article info abstract Article history: Received 13 August 2009 Available online 15 April 2010 Keywords: Critical group De Bruijn graph Iterated line digraph Kautz graph Matrix-tree theorem Oriented spanning tree Weighted Laplacian We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph G and its directed line graph LG. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at afixedvertex.inthecasewheng is regular of degree k, weshow that the sandpile group of G is isomorphic to the quotient of the sandpile group of LG by its k-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs Elsevier Inc. All rights reserved. 1. Introduction Let G = V, E be a finite directed graph, which may have loops and multiple edges. Each edge e E is directed from its source vertex se to its target vertex te. Thedirected line graph LG = E, E 2 has as vertices the edges of G, andasedgestheset E 2 = { e 1, e 2 E E se2 = te 1 }. For example, if G has just one vertex and n loops, then LG is the complete directed graph on n vertices which includes a loop at each vertex. If G has two vertices and no loops, then LG is a bidirected complete bipartite graph. An oriented spanning tree of G is a subgraph containing all of the vertices of G, havingnodirected cycles, in which one vertex, the root, has outdegree 0, and every other vertex has outdegree 1. The number κg of oriented spanning trees of G is sometimes called the complexity of G. address: levine@math.mit.edu. URL: 1 The author is supported by an NSF postdoctoral fellowship /$ see front matter 2010 Elsevier Inc. All rights reserved. doi: /j.jcta

2 L. Levine / Journal of Combinatorial Theory, Series A Our first result relates the numbers κlg and κg. Let{x e } e E and {x v } v V and consider the polynomials κ edge G, x = x e, T e T κ vertex G, x = x te. T e T The sums are over all oriented spanning trees T of G. Write be indeterminates, indegv = # { e E te = v }, outdegv = # { e E se = v } for the indegree and outdegree of vertex v in G. Wesaythatv is a source if indegv = 0. Theorem 1.1. Let G = V, E be a finite directed graph with no sources. Then κ vertex LG, x = κ edge G, x v V se=v x e indegv 1. 1 Note that since the vertex set of LG coincides with the edge set of G, both sides of 1 are polynomials in the same set of variables {x e } e E.Settingallx e = 1yieldstheproductformula κlg = κg v V outdegv indegv 1 2 due in a slightly different form to Knuth [11]. Special cases of 2 include Cayley s formula n n 1 for the number of rooted spanning trees of the complete graph K n,aswellastheformulam + nm n 1 n m 1 for the number of rooted spanning trees of the complete bipartite graph K m,n.thesearerespectively the cases that G has just one vertex with n loops, or G has just two vertices a and b with m edges directed from a to b and n edges directed from b to a. Suppose now that G is strongly connected, thatis,foranyv, w V there are directed paths in G from v to w and from w to v. Thenassociatedtoanyvertexv of G is an abelian group K G, v,the sandpile group, whoseorderisthenumberoforientedspanningtreesofg rooted at v.itsdefinition and basic properties are reviewed in Section 3. Other common names for this group are the critical group, Picard group, Jacobian, and group of components. In the case when G is Eulerian that is, indegv = outdegv for all vertices v the groups K G, v and K G, v are isomorphic for any v, v V,andweoftendenotethesandpilegroupjustbyKG. When G is Eulerian, we show that there is a natural map from the sandpile group of LG to the sandpile group of G, descendingfromthez-linear map φ : Z E Z V which sends e te. Let k be a positive integer. We say that G is balanced k-regular if indegv = outdegv = k for every vertex v. Theorem 1.2. Let G = V, E be a strongly connected Eulerian directed graph, fix e E and let v = te. The map φ descends to a surjective group homomorphism φ : KLG, e KG, v. Moreover, if G is balanced k-regular, then ker φ is the k-torsion subgroup of K LG, e.

3 352 L. Levine / Journal of Combinatorial Theory, Series A This result extends to directed graphs some of the recent work of Berget, Manion, Maxwell, Potechin and Reiner [1] on undirected line graphs. If G = V, E is an undirected graph, the undirected line graph lineg of G has vertex set E and edge set {{ e, e } e, e E, e e }. The results of [1] relate the sandpile groups of G and lineg. The undirected case is considerably more subtle, because although there is still a natural map K line G K G when G is regular, this map may fail to be surjective. A particularly interesting family of directed line graphs are the de Bruijn graphs DB n,definedrecursively by DB n = LDB n 1, n 1, where DB 0 is the graph with just one vertex and two loops. The 2 n vertices of DB n can be identified with binary words b 1...b n of length n; twosuchsequencesb and b are joined by a directed edge b, b if and only if b i = b i+1 for all i = 1,...,n 1. Using Theorem 1.2, we obtain the full structure of the sandpile groups of the de Bruijn graphs. Theorem 1.3. n 1 KDB n = Z/2 j Z 2 n 1 j. j=1 and Closely related to the de Bruijn graphs are the Kautz graphs, defined by Kautz 1 = {1, 2, 3}, { 1, 2, 1, 3, 2, 1, 2, 3, 3, 1, 3, 2 } Kautz n = LKautz n 1, n 2. The Kautz graphs are useful in network design because they have close to the maximum possible number of vertices given their diameter and degree [8] and because they contain many short vertexdisjoint paths between any pair of vertices [6]. The following result gives the sandpile group of Kautz n. Theorem 1.4. KKautz n = Z/3Z Z/2 n 1 Z n 2 2 Z/2 j Z 3 2 n 2 j. j=1 The remainder of the paper is organized as follows. In Section 2, we prove Theorem 1.1 and state a variant enumerating spanning trees with a fixed root. Section 3 begins by defining the sandpile group, and moves on from there to the proof of Theorem 1.2. In Section 4 we enumerate spanning trees of iterated line digraphs. Huaxiao, Fuji and Qiongxiang [10] prove that for a balanced k-regular directed graph G on N vertices, κ L n G = κgk kn 1N. Theorem 4.1 generalizes this formula to an arbitrary directed graph G having no sources. This section also contains the proofs of Theorems 1.3 and 1.4. Lastly, in Section 5 we pose two questions for future study.

4 L. Levine / Journal of Combinatorial Theory, Series A Spanning trees Let G = V, E be a finite directed graph, loops and multiple edges allowed. We denote its vertices by v, w,... and edges by e, f,...eachedgee E is directed from its source se to its target te. In this section we prove Theorem 1.1 relating the spanning trees of G and LG, anddiscusssome interesting special cases. If k is a field, we write k V and k E for the k-vector spaces with bases indexed by V and E respectively. We think of the elements of k V or k E as formal k-linear combinations of vertices or of edges. Consider the field of rational functions Qx = Qx e e E,x v v V. The edge-weighted Laplacian and vertex-weighted Laplacian of G are the Qx-linear transformations sending edge, vertex : Qx V Qx V edge v = se=v vertex v = se=v x e te v ; x te te v. The sums are over all edges e E such that se = v. We will use the following form of the matrix-tree theorem for directed graphs. Here [t]pt denotes the coefficient of t in the polynomial pt. Theorem 2.1 Matrix-tree theorem. κ edge G, x =[t] det t Id edge, κ vertex G, x =[t] det t Id vertex. For a proof, see for example [4, Theorem 2] for the vertex-weighted version, and [3] for the edgeweighted version. Proof of Theorem 1.1. Consider the V E matrix { 1, v = te, A ve = 0, else and the E V matrix { xe, v = se, B ev = 0, else. Let be the edge-weighted Laplacian of G, andlet L be the vertex-weighted Laplacian of LG. Then and = AB D L = BA D L 3 where D and D L are the diagonal matrices with diagonal entries D vv = s f =v x f, v V

5 354 L. Levine / Journal of Combinatorial Theory, Series A and D L ee = x f, e E. s f =te Since AD L = DA,wehave A L = A BA D L = ABA DA= AB DA = A. 4 In particular, L kera kera, sothevectorspacedecomposition Qx E = kera kera exhibits L in block triangular form. Hence the characteristic polynomial χt of L factors as χt = χ 1 tχ 2 t where χ 1 and χ 2 are respectively the characteristic polynomials of L kera and L kera. By hypothesis, G has no sources, so A has full rank. In particular, AA T is invertible. Hence the restriction A kera is an isomorphism of kera = ImA T onto Qx V. By 4 it follows that L kera and have the same characteristic polynomial χ 2 t = dett Id. Since the rows of sum to zero, χ 2 t has no constant term. By the matrix-tree theorem, κ vertex LG, x =[t]χt = χ 1 0 [t]χ 2 t = det L kera κ edge G, x. It remains to find the determinant of L kera.foreachvertexv V,fixanedgee 0 v with te 0 v = v. ThenabasisforkerA is given by the vectors α e = e e 0 v, v V, e E, te = v, e e 0 v. By 3 we have L α e = s f =te x f α e so the vectors α e form an eigenbasis for L kera. As each eigenvalue s f =v x f occurs with multiplicity indegv 1, we conclude that det L indegv 1 kera = x f. v V s f =v We remark that the idea of using the incidence matrices A and B to relate the adjacency matrices of G and LG has appeared before. See, for example, Yan and Zhang [18, Proposition 1.4], who in turn cite Lin and Zhang [12] and Liu [13]. Theorem 1.1 enumerates all oriented spanning trees of LG, whileinmanyapplicationsonewants to enumerate spanning trees with a fixed root. Given a vertex v V,let κ edge G, v, x = x e roott =v e T

6 L. Levine / Journal of Combinatorial Theory, Series A and κ vertex G, v, x = x te. roott =v e T We will use the following variant of the matrix-tree theorem; see [3] and [17, Theorem 5.6.4]. Theorem 2.2 Matrix-tree theorem, rooted version. Let edge 0 and vertex 0 be the submatrices of edge and vertex omitting row and column v.then κ edge G, v, x = det edge 0, κ vertex G, v, x = det vertex 0. The following variant of Theorem 1.1 enumerates spanning trees of LG with a fixed root e in terms of spanning trees of G with root w = se. Theorem 2.3. Let G = V, E be a finite directed graph, and let e = w, v be an edge of G. If indegv 1 for all vertices v V,andindegv 2,then κ vertex LG, e, x indegv 2 x e κ edge G, w, x = indegv 1 x e x e. se=v v v se=v Proof. The proof is analogous to that of Theorem 1.1, except that it uses reduced incidence matrices and A 0 : Qx E {e } Qx V B 0 : Qx V Qx E {e }. The edge-weighted Laplacian of the graph G \ e = V, E {e } is given by G\e = A 0 B 0 D + M where the matrix M has a single nonzero entry x e in row and column w.expandingdetd A 0 B 0 along column w we find detd A 0 B 0 = det G\e + x e det 0 where 0 is the submatrix of the edge-weighted Laplacian of G omitting the row and column w. By Theorem 2.2 we have det 0 = κ edge G, w, x. Sincetherowsof G\e sum to zero, it follows that detd A 0 B 0 = x e κ edge G, w, x. The submatrix L 0 of the vertex-weighted Laplacian of LG omitting the row and column e equals B 0 A 0 D L 0,whereDL 0 is the submatrix of DL omitting row and column e.sincea 0 D L 0 = DA 0,we have A 0 L 0 = 0 A B0 A 0 D L 0 = A0 B 0 A 0 DA 0 = A 0 B 0 DA 0 hence L 0 kera 0 kera 0.NowbyTheorem2.2, κ vertex LG, e, x = det L 0 = det L 0 kera0 det L 0 kera0.

7 356 L. Levine / Journal of Combinatorial Theory, Series A By hypothesis, the graph G \ e has no sources, so A 0 has full rank. The rest of the proof proceeds as before, giving and det L 0 kera0 = detd A0 B 0 = x e κ edge G, w, x det L 0 kera0 indegv 2 indegv 1 = x e. x e se=v v v se=v Setting all x e = 1inTheorem2.3yieldstheenumeration κlg, e = κg, w πg 5 outdegv where κg, w is the number of oriented spanning trees of G rooted at w,and πg = v V outdegv indegv 1. It is interesting to compare this formula to the theorem of Knuth [11], which in our notation reads κlg, e = κg, v 1 outdegv te=v e e κ G, se πg. 6 To see directly why the right sides of 5 and 6 are equal, we define a unicycle to be a spanning subgraph of G which contains a unique directed cycle, and in which every vertex has outdegree 1. If vertex v is on the unique cycle of a unicycle U,wesaythatU goes through v. Lemma 2.4. κ edge G, v, x x e = κ edge G, se, x xe. se=v te=v Proof. Removing e gives a bijection from unicycles containing a fixed edge e to spanning trees rooted at se. IfU is a unicycle through v,thenthecycleofu contains a unique edge e with se = v and a unique edge e with te = v,sobothsidesareequalto U e U x e where the sum is over all unicycles U through v. Setting all x e = 1inLemma2.4yields κg, v outdegv = κ G, se. te=v Hence the factor appearing in front of πg in Knuth s formula 6 is equal to κg, w /outdegv. We conclude this section by discussing some special cases and interesting examples of Theorem 1.1.

8 L. Levine / Journal of Combinatorial Theory, Series A Deletion and contraction Fix an edge e E which is not a loop, i.e., se te. Let G \ e = V, E {e} be the graph obtained by deleting e from G. While there is more than one sensible way to define contraction for directed graphs, the following definition is natural from the point of view of oriented spanning trees. Let G/e be the graph obtained from G by first deleting all edges f with s f = se, and then identifying the vertices se and te. Formally,G/e = V /e, E/e, where and V /e = V { se, te } {e} E/e = E { f s f = se }. The source and target maps for G/e are given by p s i and p t i, wherei : E/e E is inclusion, and p : V V /e is given by pse = pte = e, andpv = v for v se, te. With these definitions, the spanning tree enumerator κ edge satisfies the following deletioncontraction recurrence. Lemma 2.5. Let G be a finite directed graph, and let e be a non-loop edge of G. Then κ edge G, x = κ edge G \ e, x + x e κ edge G/e, x. Proof. Oriented spanning trees of G \ e are in bijection with oriented spanning trees of G that do not contain the edge e. WiththeabovedefinitionofG/e, oneeasilychecksthatthemapt T {e} defines a bijection from oriented spanning trees of G/e to oriented spanning trees of G that contain the edge e. Suppose now that we set x f = 1 for all f e. Thecoefficient of x l e in κ vertex LG, x then counts the number of oriented spanning trees T of LG with indeg T e = l. Ifv = se has indegree k and outdegree m, thenbytheorem1.1andlemma2.5,thisnumberisgivenbythecoefficient of x l e in the product [ κg \ e + xe κg/e ] m 1 + x e k 1 outdegw indegw 1. w v Using the binomial theorem, we obtain the following. Proposition 2.6. Let G = V, E be a finite directed graph with no sources. Fix a non-loop edge e E and an integer l 0. The number of oriented spanning trees T of LG satisfyingindeg T e = l is given by outdegw indegw 1 k 1 κg \ em 1 k 1 l k 1 + κg/em 1 k l l l 1 w v where v = se,k= indegv and m = outdegv Complete graph Taking G to be the graph with one vertex and n loops, so that LG is the complete directed graph Kn on n vertices including a loop at each vertex, we obtain from Theorem 1.1 the classical formula κ vertex Kn = x 1 + +x n n 1. For a generalization to forests, see [17, Theorem 5.3.4]. Note that oriented spanning trees of Kn are in bijection with rooted spanning trees of the complete undirected graph K n,byforgettingorientation.

9 358 L. Levine / Journal of Combinatorial Theory, Series A Complete bipartite graph Taking G to have two vertices, a and b, withm edges directed from a to b and n edges directed from b to a, weobtainfromtheorem1.1 κ vertex Km,n = x 1 + +x m + y 1 + +y n x 1 + +x m n 1 y 1 + +y n m 1, where Km,n = LG is the bidirected complete bipartite graph on m+n vertices. The variables x 1,...,x m correspond to vertices in the first part, and y 1,...,y n correspond to vertices in the second part. As with the complete graph, oriented spanning trees of Km,n are in bijection with rooted spanning trees of the undirected complete bipartite graph K m,n by forgetting orientation De Bruijn graphs The spanning tree enumerators for the first few de Bruijn graphs are κ vertex DB 1 = x 0 + x 1 ; κ vertex DB 2 = x 00 + x 01 x 10 + x 11 x 01 + x 10 ; κ vertex DB 3 = x x 001 x x 011 x x 101 x x Sandpile groups x 011 x 110 x x 010 x 110 x x 110 x 101 x x 110 x 100 x x 100 x 001 x x 101 x 001 x x 001 x 010 x x 001 x 011 x 110. Let G = V, E be a strongly connected finite directed graph, loops and multiple edges allowed. Consider the free abelian group Z V generated by the vertices of G; wethinkofitselementsasformal linear combinations of vertices with integer coefficients. For v V let v = te v Z V se=v where the sum is over all edges e E such that se = v. Fixing a vertex v V, let L V be the subgroup of Z V generated by v and { v } v v.thesandpile group K G, v is defined as the quotient group KG, v = Z V /L V. The V V integer matrix whose column vectors are { v } v V is called the Laplacian of G. By Theorem 2.2, its principal minor omitting the row and column corresponding to v counts the number κg, v of oriented spanning trees of G rooted at v.sincethisminorisalsotheindexofl V in Z V, we have #KG, v = κg, v. Recall that G is Eulerian if indegv = outdegv for every vertex v. IfG is Eulerian, then the groups K G, v and K G, v are isomorphic for any vertices v and v [9, Lemma 4.12]. In this case we usually denote the sandpile group just by K G. The sandpile group arose independently in several fields, including arithmetic geometry [14,15], statistical physics [5] and algebraic combinatorics [2]. Often it is defined for an undirected graph G; to translate this definition into the present setting of directed graphs, replace each undirected edge by a pair of directed edges oriented in opposite directions. Sandpiles on directed graphs were first studied in [16]. For a survey of the basic properties of sandpile groups of directed graphs and their proofs, see [9].

10 L. Levine / Journal of Combinatorial Theory, Series A The goal of this section is to relate the sandpile groups of an Eulerian graph G and its directed line graph LG. Tothatend,letZ E be the free abelian group generated by the edges of G. Fore E let e = f e Z E. s f =te Fix an edge e E, andletv = te.letl E Z E be the subgroup generated by e and { e } e e. Then the sandpile group associated to LG and e is KLG, e = Z E /L E. Note that LG may not be Eulerian even when G is Eulerian. For example, if G is a bidirected graph i.e., a directed graph obtained by replacing each edge of an undirected graph by a pair of oppositely oriented directed edges then G is Eulerian, but LG is not Eulerian unless all vertices of G have the same degree. We will work with maps φ and ψ relating the sandpile groups of G and LG. These maps are analogous to the incidence matrices A and B from Section 2, except that now we work over Z instead of the field Qx. Lemma 3.1. Let φ : Z E Z V be the Z-linear map sending e te. IfGisEulerian,thenφ descends to a surjective group homomorphism φ : KLG, e KG, v. Proof. To show that φ descends, it suffices to show that φl E L V.Foranye E, wehave φ e = t f te = te. s f =te The right side lies in L V by definition if te v.moreover,sinceg is Eulerian, te se = indegv outdegv v = 0, v V v = e E v V so v = v v v also lies in L V.Finally,φe = v L V,andhenceφL E L V. Since G is strongly connected, every vertex has at least one incoming edge, so φ is surjective, and hence φ is surjective. Let k be a positive integer. We say that G is balanced k-regular if indegv = outdegv = k for every vertex v. Notethatanybalancedk-regular graph is Eulerian; and if G is balanced k-regular, then its directed line graph LG is also balanced k-regular. In particular, this implies e = 0 e E so that e L E. Now consider the Z-linear map ψ : Z V Z E sending v se=v e. For a group Γ,writekΓ ={kg g Γ }. Lemma 3.2. If G is balanced k-regular, then ψ descends to a group isomorphism ψ : KG kklg.

11 360 L. Levine / Journal of Combinatorial Theory, Series A Proof. We have ψv = e + ke L E and for any vertex v V, ψ v = ψ te kψv se=v = se=v s f =te = se=v = se=v s f =te e. f k f ke sg=v g Since LG is Eulerian, the right side lies in L E.HenceψL V L E,andψ descends to a group homomorphism ψ : KG KLG. If v is any vertex of G, ande is any edge with te = v, then ψv = ke + e, so the image of ψ is kklg. To complete the proof it suffices to show that ψ 1 L E L V,sothat ψ is injective. If k = 1then K G is the trivial group, so there is nothing to prove. Assume now that k 2. Given η Z V with ψη L E,write ψη = e E b e e + b e for some coefficients b e, b Z. Then ψη b e = b e f ke e E s f =te = b e f kb e e f E te=s f e E = b e kb f f. f E te=s f Now writing η = v V a v v, sothatψη = f E a s f f, equating coefficients of f gives kb f = b e a s f, f e. 7 te=s f Note that the right side depends only on s f. Forv V,let F v = 1 k b e 1 k a v. te=v

12 L. Levine / Journal of Combinatorial Theory, Series A Then b f = F s f for all edges f e.sincek 2, for any v V there exists an edge f e with s f = v. Moreoverifv v and te = v, thene e.from7weobtain a v = b e kb f = F se kfv, v v. te=v te=v Hence η a v v = a v v = F se te kfvv v v e E, te v v v = F v te kv + kfv v v V se=v, te v = F v v + kfv F se v. v V te=v The right side lies in L V,soη L V,completingtheproof. Proof of Theorem 1.2. If G is Eulerian, then φ descends to a surjective homomorphism of sandpile groups by Lemma 3.1. If G is balanced k-regular, then ψ is injective by Lemma 3.2, so ker φ = ker ψ φ. Moreover for any edge e E ψ φe = s f =te f = ke + e. Hence ψ φ is multiplication by k, andker φ is the k-torsion subgroup of K LG. 4. Iterated line graphs Let G = V, E be a finite directed graph, loops and multiple edges allowed. The iterated line digraph L n G = E n, E n+1 has as vertices the set E n = { e 1,...,e n E n sei+1 = te i, i = 1,...,n 1 } of directed paths of n edges in G. TheedgesetofL n G is E n+1,andtheincidenceisdefinedby se 1,...,e n+1 = e 1,...,e n ; te 1,...,e n+1 = e 2,...,e n+1. We also set E 0 = V,andL 0 G = G. For example, the de Bruijn graph DB n is L n DB 0,whereDB 0 is the graph with one vertex and two loops. Our next result relates the number of spanning trees of G and L n G.Givenavertexv V,let pn, v = # { e 1,...,e n E n te n = v } be the number of directed paths of n edges in G ending at vertex v. Theorem 4.1. Let G = V, E be a finite directed graph with no sources. Then κ L n G = κg v V outdegv pn,v 1.

13 362 L. Levine / Journal of Combinatorial Theory, Series A Proof. For any j 0, by Theorem 1.1 applied to L j G with all edge weights 1, κl j+1 G κl j G = outdeg te j indegse 1 1 e 1,...,e j E j = v V outdegv p j+1,v p j,v. Taking the product over j = 0,...,n 1yieldstheresult. When G is balanced k-regular, we have pn, v = k n for all vertices v, soweobtainasaspecial case of Theorem 4.1 the result of Huaxiao, Fuji and Qiongxiang [10, Theorem 1] κ L n G = κgk kn 1#V. In particular, taking G = DB 0 yields the classical formula κdb n = 2 2n 1. Since DB n is Eulerian, the number κdb n, v of oriented spanning trees rooted at v does not depend on v,so κdb n, v = 2 n κdb n = 2 2n n 1. 8 This familiar number counts de Bruijn sequences of order n + 1EuleriantoursofDB n uptocyclic equivalence. De Bruijn sequences are in bijection with oriented spanning trees of DB n rooted at a fixed vertex v ;formoreontheconnectionbetweenspanningtreesandeuleriantours,see[7]and[17, Section 5.6]. Perhaps less familiar is the situation when G is not regular. As an example, consider the graph G = {0, 1}, { 0, 0, 0, 1, 1, 0 }. The vertices of its iterated line graph L n G are binary words of length n + 1containingnotwoconsecutive 1 s. The number of such words is the Fibonacci number F n+3,andthenumberofwordsending in 0 is F n+2.bytheorem4.1,thenumberoforientedspanningtreesofl n G is κ L n G = 2 2 pn,0 1 = 2 F n+2. Next we turn to the proofs of Theorems 1.3 and 1.4. If a and b are positive integers, we write Z a b for the group Z/bZ Z/bZ with a summands. Proof of Theorem 1.3. Induct on n. From8wehave hence #KDB n = 2 2n n 1 KDB n = Z a 1 2 Za 2 4 Za 3 8 Za m 2 m for some nonnegative integers m and a 1,...,a m satisfying m ja j = 2 n n 1. 9 j=1 By Lemma 3.2 and the inductive hypothesis,

14 L. Levine / Journal of Combinatorial Theory, Series A Z a 2 2 Za 3 4 Za m 2 m 1 2KDB n hence m = n 1and KDB n 1 a 2 = 2 n 3, a 3 = 2 n 4,...,a n 1 = 1. Z 2n 3 2 Z 2n 4 4 Z 2 n 2 Solving 9 for a 1 now yields a 1 = 2 n 2. For p prime, by carrying out the same argument on a general balanced p-regular directed graph G on N vertices, we find that where K L n 1 m n G K Z p j pn 1 j p 1 2N Z p n p 1N r 1 Z p n+ j a j j=1 Sylow p KG = Zp a 1 Z p m a m ; K = KG/Sylow p KG ; r = a 1 + +a m. In particular, taking G = Kautz 1 with p = 2, we have K G = K = Z3,andwearriveatTheorem Concluding remarks Theorem 1.2 describes a map from the sandpile group K LG, e to the group K G, v when G is an Eulerian directed graph and e = w, v is an edge of G. Thereisalsoasuggestivenumerical relationship between the orders of the sandpile groups K LG, e and K G, w,whichholdseven when G is not Eulerian: by Eq. 5 we have κg, w κlg, e whenever G satisfies the hypothesis of Theorem 2.3. This observation leads us to ask whether K G, w can be expressed as a subgroup or quotient group of K LG, e. The area of spanning trees, Eulerian tours, and sandpile groups is full of simple enumerative results with no known bijective proofs. To give just one example, the number of de Bruijn sequences of order n Eulerian tours of DB n 1 with distinguished starting edge is 2 2n 1.RichardStanleyhasposed the problem of finding a bijection between ordered pairs of such sequences and all 2 2n binary words of length 2 n. This problem and a number of others could be solved by giving a bijective proof of Theorem 1.1. References j=1 [1] A. Berget, A. Manion, M. Maxwell, A. Potechin, V. Reiner, The critical group of a line graph, [2] N.L. Biggs, Chip-firing and the critical group of a graph, J. Algebraic Combin [3] S. Chaiken, A combinatorial proof of the all minors matrix tree theorem, SIAM J. Algebraic Discrete Methods [4] F.R.K. Chung, R.P. Langlands, A combinatorial Laplacian with vertex weights, J. Combin. Theory Ser. A [5] D. Dhar, Self-organized critical state of sandpile automaton models, Phys. Rev. Lett [6] D.-Z. Du, Y.-D. Lyuu, F.D. Hsu, Line digraph iterations and connectivity analysis of de Bruijn and Kautz graphs, IEEE Trans. Comput [7] T. van Aardenne-Ehrenfest, N.G. de Bruijn, Circuits and trees in oriented linear graphs, Simon Stevin [8] M.A. Fiol, J.L.A. Yebra, I.A. De Miquel, Line digraph iterations and the d, k-digraph problem, IEEE Trans. Comput

15 364 L. Levine / Journal of Combinatorial Theory, Series A [9] A.E. Holroyd, L. Levine, K. Mészáros, Y. Peres, J. Propp, D.B. Wilson, Chip-firing and rotor-routing on directed graphs, in: In and Out of Equilibrium 2, in: Progr. Probab., vol. 60, Birkhäuser, 2008, pp , [10] Z. Huaxiao, Z. Fuji, H. Qiongxiang, On the number of spanning trees and Eulerian tours in iterated line digraphs, Discrete Appl. Math [11] D.E. Knuth, Oriented subtrees of an arc digraph, J. Combin. Theory [12] G.N. Lin, F.J. Zhang, The characteristic polynomial of line digraph and type of cospectral digraph, Kexuetongbao in Chinese. [13] B.L. Liu, Combinatorial Matrix Theory, Science Press, 1996 in Chinese. [14] D.J. Lorenzini, Arithmetical graphs, Math. Ann [15] D.J. Lorenzini, A finite group attached to the Laplacian of a graph, Discrete Math [16] E.R. Speer, Asymmetric abelian sandpile models, J. Stat. Phys [17] R.P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, [18] W. Yan, F. Zhang, Heredity of the index of convergence of the line digraph, Discrete Appl. Math