Cubic graphs, their Ehrhart quasi-polynomials, and a scissors congruence phenomenon
Cristina G. Fernandes, José C. de Pina, Jorge Luis Ramírez Alfonsín, Sinai Robins
aa r X i v : . [ m a t h . C O ] F e b Cubic graphs, their Ehrhart quasi-polynomials,and a scissors congruence phenomenon ∗ Cristina G. Fernandes † José C. de Pina † Jorge Luis Ramírez Alfonsín ‡ Sinai Robins † February 21, 2018
Abstract
The scissors congruence conjecture for the unimodular group is an analogue ofHilbert’s third problem, for the equidecomposability of polytopes. Liu and Osser-man studied the Ehrhart quasi-polynomials of polytopes naturally associated tographs whose vertices have degree one or three. In this paper, we prove the scissorscongruence conjecture, posed by Haase and McAllister, for this class of polytopes.The key ingredient in the proofs is the nearest neighbor interchange on graphs anda naturally arising piecewise unimodular transformation. A cubic graph is a graph whose vertices have degree three and a { , } -graph is a graphwhose vertices have degree either one or three. The graphs are allowed to have loops andparallel edges. Motivated by a result of Mochizuki [11], Liu and Osserman [10] associateda polytope P G to each { , } -graph G and studied its Ehrhart quasi-polynomial.For each degree three vertex v of a { , } -graph G = ( V, E ), let a , b , and c be the threeedges incident to v . Denote by S ( v ) the linear system consisting of a perimeter inequalityand three metric inequalities defined on the variables w a , w b , and w c as follows: w a + w b + w c ≤ w a ≤ w b + w c w b ≤ w a + w c w c ≤ w a + w b . ∗ FAPESP 13/03447-6 and 15/10323-7, CNPq 456792/2014-7 and 452507/2016-2, CAPES PROEX. † Instituto de Matemática e Estatística, Universidade de São Paulo, 05508-090 São Paulo, Brazil.( [email protected] , [email protected] , [email protected] ). ‡ IMAG, Univ. Montpellier, CNRS, Montpellier, France. ( [email protected] ). w a , w b , and w c are nonnegative. Consider the union of all the linear systems S ( v ), taken over all degreethree vertices v of G . The polytope P G is defined by the set of all real solutions for thislinear system.Let E = { , . . . , m } and w : E → R be a weight function defined on the edges of G .We use the vector notation w = ( w , . . . , w m ) ∈ R m . In particular, when w is a solutionfor the linear system defining P G , we write w ∈ P G .Given a rational polytope P , Eugène Ehrhart defined the function L P ( t ) := | t P ∩ Z m | ,which is the number of lattice points in the closed dilated polytope t P , for a nonnegativeinteger parameter t . Ehrhart showed that this function is a polynomial in t when P isan integral polytope. More generally, if P is a rational polytope, the function L P ( t ) isa quasi-polynomial whose period is closely related to the denominators appearing in thecoordinates of the vertices of P [6, 2].Liu and Osserman conjectured ([10, Conj. 4.2]) that polytopes associated to con-nected { , } -graphs with the same number of vertices and edges have the same Ehrhartquasi-polynomial. They partially proved their conjecture, by showing that these quasi-polynomials coincide for all nonnegative odd values of the dilation parameter t . In 2013,Wakabayashi [15, Thm. A(ii)] proved their conjecture.An ingredient in Wakabayashi’s proof for Liu and Osserman’s conjecture is a localtransformation performed in { , } -graphs. This transformation is called an A -move byWakabayashi and it is also known as a nearest neighbor interchange (NNI). The NNI hasbeen studied mainly for binary trees [4, 12], cubic graphs [14], and { , } -graphs [15]. Wepresent the following general result for connected graphs with the same degree sequence,which might of interest on its own. We refer to a vertex of degree one in a graph simplyas a leaf . An edge is external if it is incident to a leaf, otherwise it is internal . Theorem 1.
Let G and G ′ be connected graphs with the same degree sequence and thesame set of external edges. Then(a) G can be transformed into G ′ through a series of NNI moves.(b) One can choose a spanning tree in G and a spanning tree in G ′ and require that allthe pivots of the NNI moves are internal edges of both of these spanning trees. In particular, for the special case of { , } -graphs, the proof provided for Theorem 1(a)constitutes an alternative graph theoretic proof of a proposition by Wakabayashi [15,Prop. 6.2].One of the concerns of this paper is on the scissors congruence conjecture for the uni-modular group, which is an analogue of Hilbert’s third problem (equidecomposability).Concretely, this was stated as the following question by Haase and McAllister [7]. Anintegral matrix U is unimodular if it has determinant ±
1. An affine unimodular trans-formation is defined by x → U x + b , where U is a unimodular matrix and b is a realvector. 2 uestion 2. [7, Question 4.1] Suppose that P and P ′ are polytopes with the same Ehrhartquasi-polynomial. Is it true that there is a decomposition of P into relatively open simplices P , . . . , P k and affine unimodular transformations U , . . . , U k such that P ′ is the disjointunion of U ( P ) , . . . , U r ( P r ) ? We show that for polytopes associated to { , } -graphs such a scissors congruencedecomposition holds. Namely, we have the following. Theorem 3.
Let G and G ′ be two connected { , } -graphs with the same number ofvertices and edges. Then there is a dissection of P G into smaller polytopes P G , . . . , P kG and affine unimodular transformations U , . . . , U k such that P G ′ is the union of U ( Q G ) , . . . , U k ( Q kG ) . The proof of Theorem 3 relies on a piecewise unimodular transformation associatedto a weighted version of the NNI move.The rational polytope P G , associated to a { , } -graph G , enjoys some fascinatingsymmetry. Linke [9] considered the extension of L P ( t ) for all nonnegative real numbers t .Royer [13] defined a polytope to be semi-reflexive if L P ( s ) = L P ( ⌊ s ⌋ ) for every nonnegativereal number s . One can verify that P G is semi-reflexive. A polytope is reflexive if it isintegral, the origin is in its interior, and it is semi-reflexive [2]. We prove the following. Theorem 4.
For each { , } -graph G , the polytope P G − is reflexive. The paper is organized as follows. Section 2 contains the results involving the NNImove in graphs with the same degree sequence, including Theorem 1, while Section 3discusses the extension of the NNI move to weighted graphs. Section 4 describes theunimodular decomposition of P G and presents the proof of Theorem 3. In Section 5, weprove Theorem 4. Finally, Section 6 contains some concluding remarks. With an eye towards Section 3, where we consider weighted graphs, we think of a graphas defined by Bondy and Murty [3]. A graph G is an ordered pair ( V, E ) consisting ofa set V of vertices and a set E , disjoint from V , of edges, together with an incidencefunction ψ G that associates with each edge of G an unordered pair of (not necessarilydistinct) vertices of G . The degree sequence of G is the monotonic nonincreasing sequenceconsisting of the degrees of its vertices.A nearest neighbor interchange (NNI) is a local move performed in G on a trail W oflength three. This move interchanges one end of the two extreme edges of W on the centraledge (Figure 1). We refer to the central edge of W as the pivot of the NNI move. Theresult of the move is another graph G ′ on the same number of connected components,with the same degree sequence. We consider the graph G ′ as having the same set of3ertices and edges, that is, G ′ = ( V, E ), and only the incidence function ψ G ′ is adjustedaccordingly. Intuitively, one can think of the edges as sticks that are being moved from G to G ′ . NNI b b b b a b a b Figure 1: An NNI move on the trail marked in bold. One of the incidences of the edges a and b were interchanged.We think of an NNI move as a function γ W that associates the graph G to the graph G ′ .In symbols, γ W ( G ) = G ′ . Observe that W is also a trail in G ′ and γ W ( G ′ ) = G .The well-known rotation, used in data structures to balance binary trees, is a particularNNI move, performed on a { , } -tree (Figure 2). left rotationright rotation r cba r cba b b b b Figure 2: Right and left rotations applied to a binary tree.An NNI move does not affect the incidence to leaves, thus it preserves the partitionof the edge set into internal and external edges.Culik and Wood [4, Thm. 2.4] proved that any two { , } -trees with ℓ (labelled)leaves can be transformed into one another through a finite series of NNI moves. Theyadditionally gave an upper bound of 4 ℓ −
12 + 4 ℓ log ( ℓ/
3) on the number of NNI movesneeded for this transformation. In this section, first we extend Culik and Wood’s theoremto trees with the same degree sequence (Lemma 8), which we then use to extend theirresult further to connected graphs with the same degree sequence (Theorem 1).A caterpillar is a tree for which the removal of all leaves results in a path, called its central path , or in the empty graph. For the later, we define that the central path isempty.
Lemma 5.
Any tree can be transformed into a caterpillar with the same degree sequencethrough a series of NNI moves.Proof.
Let T be a tree. If T is a caterpillar, there is nothing to prove. So, we mayassume T is not a caterpillar. Let P be a longest path in T and uv an internal edge of T not in P such that u is a vertex in P . The vertex u has two neighbors in P , otherwise P v is not in P . Let w and z be the two neighbors of u in P .Let W be a trail with edges wu , uv , and vv ′ , where v ′ is a neighbor of v other than u .Perform an NNI on the trail W as in Figure 3, to insert v in P , obtaining another tree T ′ with the same degree sequence and a path longer than P . By repeating this process, weobtain a desired caterpillar after a finite number of NNI moves. vuw z w zvu zuvw b bbb b b b bbb b b b bb Figure 3: An NNI to insert v in the central path.The spine of a caterpillar is the sequence of the degrees of the vertices in the centralpath. We say the caterpillar is ordered if its spine is a monotonic nonincreasing sequence(Figure 4).(a) bbbb b b b b (b)Figure 4: (a) A caterpillar with spine (4 , , ,
3) and central path highlighted in bold.(b) An ordered caterpillar.
Lemma 6.
Any caterpillar can be transformed into an ordered caterpillar with the samedegree sequence through a series of NNI moves.Proof.
An NNI can be used to swap any two adjacent vertices in the central path of acaterpillar, as in Figure 5(a). So we use an NNI to decrease, one by one, the number ofinversions in the spine of a caterpillar until we obtain an ordered caterpillar.
Lemma 7.
The external edges of a caterpillar can be sorted arbitrarily through a seriesof NNI moves.Proof.
An NNI can be used to swap any two external edges incident to two adjacentvertices in the central path of a caterpillar, as in Figure 5(b).The next lemma is an extension of previous works in the literature [4, 12, 14, 15] thatmight be of independent interest. 5 vuw zvuw v u zw (a) (b) deac b zvuw dcba e zvuw aedc b zvuw bbb bbb b bbbb b bb b bbbbbbbbb
Figure 5: (a) An NNI to swap two adjacent vertices in the central path of a caterpillar.(b) NNIs to swap labels on leaves hanging from adjacent vertices in the central path.
Lemma 8.
Any two trees with the same degree sequence and the same set of externaledges can be transformed into one another through a series of NNI moves.Proof.
Let T and T ′ be two trees with the same degree sequence and the same set ofexternal edges. Using Lemmas 5 and 6, we obtain a series of NNI moves that transforms T into an ordered caterpillar with the same degree sequence and the same set of externaledges of T . Similarly, we obtain another series for T ′ . Using Lemma 7, we extend theseries of NNI moves for T to sort the external edges of the caterpillar coming from T into the order they appear in the caterpillar coming from T ′ . The composition of thesetwo series of NNI moves, with the series for T ′ inverted, gives a series of NNI moves thattransforms T into T ′ .Let G be a connected graph that is not a tree, and let e be an edge that is in a cycle.The graph obtained from G by cutting e is the graph G ′ resulting from the splitting of e into two edges, each connecting one of the ends of e to one of two new leaves (Figure 6). eG b bbb b b bb bb G ′ Figure 6: Graph G ′ obtained from G by cutting an edge e .6 roof of Theorem 1. Let n be the number of vertices and m be the number edges in G and G ′ . The proof of the theorem is by induction on the dimension r = m − n + 1 of thecycle space of G and G ′ . If r = 0, then G and G ′ are trees and the theorem follows fromLemma 8. Hence we may assume that r > G and an edge in a cycle of G ′ . Let us denote thesetwo edges by e . Let H and H ′ be the graphs obtained from G and G ′ , respectively, bycutting their edge e . The number of vertices in H and H ′ is n + 2 and the number ofedges is m + 1. Call e ′ and e ′′ the two new edges in both H and H ′ . Since e is in acycle in G and in a cycle in G ′ , the graphs H and H ′ are connected and the dimensionof their cycle space is r −
1. Also, H and H ′ have the same degree sequence and thesame set of external edges. By induction, H can be transformed into H ′ through a seriesof NNI moves. The same series of NNI moves transforms G into G ′ . Indeed, the set ofexternal edges in all graphs obtained during the application of this series of NNI moves,transforming H into H ′ , contain e ′ and e ′′ . Glue e ′ and e ′′ into an edge in each of thesegraphs, obtaining a sequence of connected graphs that is the result of a series of NNImoves starting at G and ending at G ′ . This ends the proof of (4). G G ⇓ cut ⇓ cut H = H NNI ←→ · · · NNI ←→ H k = H ′ ⇓ glue ⇓ glue ⇓ glue G = G NNI ←→ · · · NNI ←→ G k = G ′ Similarly, by induction in the cycle space of G and G ′ , one can prove (b). Indeed, it issufficient to choose in each step of the induction an edge to be cut which is not in eitherof the spanning trees.For the case of cubic graphs, the proof of Theorem 1 provides an alternative prooffor a theorem by Tsukui [14, Thm. II], which refers to NNI moves as ˜ S -transformations,where the ‘S’ stands for slide . For the slightly more general case of { , } -graphs, theproof of Theorem 1 provides an alternative proof for a proposition by Wakabayashi [15,Prop. 6.2], which refers to NNI moves as A -moves. Wakabayashi’s proof uses a topologicalpants decomposition for compact, oriented surfaces of finite genus.Figure 7 shows a series of NNI moves from the complete graph K to a tree with aloop added to each of its leaves. { , } -graphs To deal with weights on the edges of a { , } -graph, we now enhance the NNI move. Thiswas achieved by a bijection defined by Wakabayashi [15, Prop. 6.3].Let G = ( V, E ) be a { , } -graph and w be a weight function defined on the edgesof G . A weighted NNI is a local move performed in ( G, w ) on a trail W of length three7 b bbbbb b bbb bbbb bbbb bbbb b +2+1 − − − − − − − − − − − − − − − − − − bb bbbbb b K T Figure 7: A series of NNI moves from K to the graph T .8nduced by a NNI move in G on W . The result of the move is the graph G ′ obtainedfrom G by applying an NNI move on W , and the weight function w ′ defined on the edgesof G ′ as follows.Let e be the central edge of W , that is, the pivot of the NNI move. Let a and b bethe other edges in W , and c and d be the remaining edges adjacent to e , as depicted inFigure 8. Possibly a , b , c , and d are not pairwise distinct. The weight function w ′ is suchthat w ′ f = w f for every f = e and w ′ e = w e + max { w a + w c , w b + w d } − max { w b + w c , w a + w d } . Note that, if w is integer valued, then so is w ′ . Moreover, since pivots are always internaledges and an NNI move does not affect the partition of the edges into internal and external,a weighted NNI move may only change the weights of internal edges. w e w b w d w a w c w ′ e w b w d w a w c b b b b G G ′ wNNIFigure 8: Edges and weights in a weighted nearest neighbor interchange.We think of a weighted NNI move as a function φ W ( G, w ) = ( G ′ , w ′ ) , which extendsthe previously defined NNI move, γ W ( G ) = G ′ . Note that φ W ( G ′ , w ′ ) = ( G, w ), because γ W ( G ′ ) = G and w ′ e + max { w ′ b + w ′ c , w ′ a + w ′ d } − max { w ′ a + w ′ c , w ′ b + w ′ d } = w ′ e + max { w b + w c , w a + w d } − max { w a + w c , w b + w d } = w e + max { w a + w c , w b + w d } − max { w b + w c , w a + w d } + max { w b + w c , w a + w d } − max { w a + w c , w b + w d } = w e . Wakabayashi [15, Prop. 6.3] proved that w ∈ t P G if and only if w ′ ∈ t P G ′ for everyinteger t ≥
0, which implies Liu and Osserman’s conjecture. We observe that this holdsalso for every real t ≥
0, and state it below as we will use it in the next section.
Lemma 9.
Let G be a { , } -graph and w be a weight function defined on the edges of G .Let W be a trail in G of length three and suppose that φ W ( G, w ) = ( G ′ , w ′ ) . Then w ∈ t P G if and only if w ′ ∈ t P G ′ for every t ≥ . Therefore there are distinct rational polytopes whose Ehrhart quasi-polynomials co-incide for all real t .As a weighted NNI changes only the weight of the pivot, which is always an internaledge, the series of NNI moves from G to G ′ changes only the weights of internal edges.9n fact, by Theorem 1(b), one can choose a spanning tree T in G and a spanning tree T ′ in G ′ and require that the series of NNI moves uses as pivots only internal edges of T and T ′ . As a consequence, the bijection from P G to P G ′ keeps fixed the majority of thecoordinates of the points. Namely, it changes only coordinates that correspond to internaledges of the chosen spanning trees. Formally, the latter discussion provides the followingas a corollary of Theorem 1(b) and Lemma 9. Let E = { , . . . , m } and w : E → R be aweight function defined on the edges of G . If X is a subset of E , then the restriction of w to X is the function w | X : X → R such that w | X ( x ) = w ( x ) for all x ∈ X . Corollary 10.
Let G and G ′ be connected { , } -graphs with the same number of verticesand on the same set E of edges. Then there exists a bijection φ between P G and P G ′ suchthat, for w ′ = φ ( w ) , we have that w ′ | X = w | X where E \ X is the set of internal edges of arbitrary spanning trees in G and G ′ . Haase and McAllister [7] have raised Question 2 that can be thought of as an analogue ofHilbert’s third problem (equidecomposability) for the unimodular group. We show thatfor polytopes associated to { , } -graphs a statement similar to Question 2 holds.Let G be a { , } -graph with edge set E = { , . . . , m } . Let W be a trail in G of length three and suppose that φ W ( G, w ) = ( G ′ , w ′ ). As argued ahead, the function φ W ( G, · ) : R m → R m is associated to one or two hyperplanes in R m and two or fourunimodular transformations. For short, let φ ( w ) = φ W ( G, w ) for every w ∈ R m . Let a , b , c , d , and e be as in Figure 8, with a , e , and b being the edges of W . Clearly φ is piecewiselinear, namely, for w ∈ R m , w ′ f = w f for every f = e and w ′ e = w e + w b − w d if w a + w b ≥ w c + w d and w a + w d ≥ w b + w c , (1a) w ′ e = w e + w a − w c if w a + w b ≥ w c + w d and w a + w d < w b + w c , (1b) w ′ e = w e + w c − w a if w a + w b < w c + w d and w a + w d ≥ w b + w c , (1c) w ′ e = w e + w d − w b if w a + w b < w c + w d and w a + w d < w b + w c . (1d)The hyperplanes associated to φ are w a + w b − w c − w d = 0 and w a − w b − w c + w d = 0,which are either the same hyperplane (if a = b or c = d ) or two orthogonal hyperplanes.Moreover, the matrix that gives the linear transformation in each case is unimodular.Indeed, the matrix for case (1a) is obtained from the identity matrix by substituting therow corresponding to the edge e by the row χ e + χ b − χ d , the matrix for case (1b) isobtained from the identity matrix by substituting the row corresponding to the edge e by the row χ e + χ a − χ c , the matrix for case (1c) is obtained from the identity matrix bysubstituting the row corresponding to the edge e by the row χ e + χ c − χ a , and the matrix10or case (1d) is obtained from the identity matrix by substituting the row correspondingto the edge e by the row χ e + χ d − χ b . Thus the determinant of each such matrix isalways 1. Therefore each of them is unimodular. bb bb (0 , , )( , , , , ) bb bb b b b ( , , )( , , b w w w w w w bb w + w + w ≤ w + w ≤ w ≤ w + w w ≤ w w ≤ w + w w ≥ w ≤ w + w w + w ≤ w ≤ w Figure 9: The polytopes of the two cubic graphs on two vertices. The shaded trianglesare the intersection of the polytopes with the hyperplane w = w .Figure 9 shows an example with the two 3-regular graphs on two vertices and theirpolytopes. The two graphs differ by one NNI move. The function φ ( w ) = w ′ between thetwo polytopes in Figure 9 is defined by w ′ = w , w ′ = w , and w ′ = w + max { w + w , w + w } − max { w , w } = w + w + w − { w , w } = ( w − w + w if w ≥ w w + w − w if w ≤ w , which is piecewise unimodular. In this case, only one hyperplane (the one containing theshaded triangle inside the polytopes, w = w ) splits the polytopes into two, each part be-ing unimodularly equivalent to one of the parts of the other polytope. The correspondingunimodular transformations are given by U w ≤ w = − −
00 1 01 − U w ≥ w = − −
00 1 0 − . Note that the polytope on the right side of Figure 9 has one more vertex than theother one, so that the combinatorial types of these piecewise unimodularly equivalent11olytopes may differ. Observe how the extra vertex ( , , ) is “formed” when going fromthe polytope on the left to the one on the right, and how it “disappears” when goingin the other direction. Also, the edge between the origin and the vertex ( , ,
0) in thepolytope on the left is not an edge in the polytope on the right.Theorem 3 extends this for graphs that differ by a series of NNI moves. It relates toQuestion 2.
Proof of Theorem 3.
The proof is constructive. By Theorem 1, there exists a finite se-quence of NNI moves that transforms G to G ′ , say G = G NNI ←→ · · · NNI ←→ G k = G ′ . Weshall explain the procedure for the first two NNI moves assuming that both underlyingweighted NNI moves are associated to only one hyperplane and consequently two uni-modular transformations. The other cases and the rest of the NNI moves follow in ananalogous but more cumbersome fashion.Let W be the trail used in the first NNI move, that goes from G = G to G , andlet φ ( w ) = φ W ( G , w ), for w ∈ R E , be the corresponding weighted NNI move. Let H bethe hyperplane associated to φ , and let U , U be the two unimodular transformations,one for each side of the hyperplane H . The hyperplane H dissects the polytope P G into two smaller polytopes, Q G and Q G , so that P G = Q G ∪ Q G . We now have that P G = U ( Q G ) ∪ U ( Q G ). In words, we presented a dissection of P G into two smallerpolytopes and two affine unimodular transformations U and U that, if applied to thetwo smaller polytopes, result in P G . Now we will proceed one more step, and present adissection of P G into four smaller polytopes, and four affine unimodular transformationsthat, if applied to the four smaller polytopes, will result in P G (Figure 10).Let W be the trail used in the second NNI move, that goes from G to G , and let φ ( w ) = φ W ( G , w ), for w ∈ R E , be the corresponding weighted NNI move. Let H bethe hyperplane associated to φ , and let T , T be the two unimodular transformations,one for each side of the hyperplane H . The hyperplane H dissects the polytope P G into two smaller polytopes, Q G and Q G , so that P G = T ( Q G ) ∪ T ( Q G ).Now, note that H dissects U ( Q G ) and U ( Q G ) each into two smaller polytopes,obtaining the dissection P G = ( Q G ∩ U ( Q G )) ∪ ( Q G ∩ U ( Q G )) ∪ ( Q G ∩ U ( Q G )) ∪ ( Q G ∩ U ( Q G )) . The latter naturally induces the following dissection: P G = U − ( Q G ∩ U ( Q G )) ∪ U − ( Q G ∩ U ( Q G )) ∪ U − ( Q G ∩ U ( Q G )) ∪ U − ( Q G ∩ U ( Q G )) . So, if we let P = U − ( Q G ∩ U ( Q G )) = U − ( Q G ) ∩ Q G , P = U − ( Q G ∩ U ( Q G )) = U − ( Q G ) ∩ Q G , P = U − ( Q G ∩ U ( Q G )) = U − ( Q G ) ∩ Q G , P = U − ( Q G ∩ U ( Q G )) = U − ( Q G ) ∩ Q G , b bbb b (0 , , , , b ( , , )( , , bb b ( , , )(0 , , )NNI on W = (1 , , W = (1 , , bb b w w w b w w w bb ( , , ) ( , , , , ) b bb b ( , , )( , , b w w w bb w w w b ( , , , , ) b bb b (0 , , )( , , , , ) bb w w w b w w w b bb NNI on W = (1 , , W = (1 , , , , )( , , ) bb b ( , , , , ) b ( , ,
0) (0 , , bbbbbb − − − U w ≤ w U w ≥ w − − − U w ≥ w − − −
10 0 1 − − −
10 0 1 U w ≤ w Figure 10: Example for Theorem 3.13hen P G = P ∪ P ∪ P ∪ P , and P G = T ( Q G ) ∪ T ( Q G )= (cid:16) T ( Q G ∩ U ( Q G )) ∪ T ( Q G ∩ U ( Q G )) (cid:17) ∪ (cid:16) T ( Q G ∩ U ( Q G )) ∪ T ( Q G ∩ U ( Q G )) (cid:17) = (cid:16) T U ( P ) ∪ T U ( P ) (cid:17) ∪ (cid:16) T U ( P ) ∪ T U ( P ) (cid:17) . This completes the proof for the two first weighted NNI moves assuming that both areassociated to only one hyperplane and consequently two unimodular transformations.For the remaining cases, whenever a weighted NNI move is associated to two hy-perplanes (and consequently four unimodular transformations), the polytopes would bedissected into up to four smaller polytopes, but the process would be essentially thesame.
An equivalent way to define a reflexive polytope is to require that it is integral and hasthe hyperplane description { x ∈ R d | Ax ≤ } for some integral matrix A . Anotherequivalent definition of reflexivity is to say that a polytope P is reflexive if and only if theorigin is in P ◦ and ( t + 1) P ◦ ∩ Z d = t P ∩ Z d for all t ∈ Z ≥ , where P ◦ is the interiorof P . Proof of Theorem 4.
The polytope 4 P G − consists of the vectors w ∈ R m satisfying w a + w b + w c ≤ w a − w b − w c ≤ − w a + w b − w c ≤ − w a − w b + w c ≤ , for each degree three vertex v and edges a , b , and c incident to v , for a total of 4 n inequal-ities, where n is the number of degree three vertices in G . From Liu and Osserman [10,Prop. 3.5], the polytope 4 P G is integral, and thus so is 4 P G − .The reflexivity of the integral polytope 4 P G − has some interesting consequences, asfollows. Let m denote the number of edges of G . Since 4 P G is an m -dimensional integralpolytope, we have that for t ≡ L P G ( t ) = t + mm ! + h ∗ t + m − m ! + · · · + h ∗ d − t + 1 m ! + h ∗ d td ! , h ∗ , . . . , h ∗ d − , h ∗ d are the coefficients of the numerator of the rational function givenby the Ehrhart series of 4 P G [2, Lemma 3.14].The polytopes 4 P G and 4 P G − have the same Ehrhart polynomial. Hence it followsfrom the reflexivity of 4 P G − and from Hibi’s palindromic theorem [2, Thm. 4.6] that h ∗ k = h ∗ m − k for all 0 ≤ k ≤ m/
2. Therefore it follows that to describe L P G ( t ), we only needto compute half of its coefficients, thus lowering the computational complexity requiredto compute L P G ( t ). In the process of proving the main theorem, a great deal of structure of the polytopes P G ,and their corresponding cones has been revealed. The polytopes P G and their corre-sponding cones may be related to the well-known metric polytopes and metric cones [5].In particular, in the case that G is a planar graph, the polytope P G can be thought of asa restricted type of “metric polytope” [8], associated to the dual graph G ∗ . It would beinteresting to pursue connections to metric polytopes.Wakabayashi [15] gave a formula for two of the four constituent polynomials of theEhrhart quasi-polynomial for P G and for the volume of P G when G is a connected cubicgraph on n vertices. It is rather surprising that the Verlinde formula makes an appearancehere, namely, Wakabayashi showed that, for odd t , L P G ( t ) = ( t + 2) n/ n +1 t +1 X j =1 n (cid:16) πjt +2 (cid:17) and vol( P G ) = | B n | n ! , where B n is the n -th Bernoulli number. Zagier [16] showed that the above Verlindeformula is the polynomial L P G ( t ) = ( t + 2) n/ n +1 n/ X k =0 ( − k − k B k (2 k )! c k ( t + 2) k , where c k is the coefficient of x − k in the Laurent expansion of sin − n x at x = 0. It wouldbe interesting to determine if a similar formula holds for all connected { , } -graphs.Even when G is a { , } -tree, it is completely open, and quite interesting, to determinethe Ehrhart quasi-polynomial of P G . We determined the following quasi-polynomials forsome small trees with LattE [1]. 15 Ehrhart polynomial P G t + t + ( t + 1 , if t is even t + , if t is odd t + t + ( t + t + t + 1 , if t is even t + t + t + , if t is odd t + t + n t + t + t + t + t + 1 , if t is even t + t + t + t + t + , if t is odd t + t + n t + t + t + t + t + t + t + 1 , if t is even t + t + t + t + t + t + t + , if t is odd Further generalizations to more general graphs and connections with other topics asmanifold invariants would also be of interest.
We thank Tiago Royer and Fabrício Caluza Machado for valuable comments.
References [1] V. Baldoni, N. Berline, J.A. De Loera, B. Dutra, M. Köppe, S. Moreinis, G. Pinto,M. Vergne, and J. Wu. Software and user’s guide for LattE integrale, October 2014. .[2] M. Beck and S. Robins.
Computing the Continuous Discretely . Springer, secondedition, 2009.[3] J.A. Bondy and U.S.R. Murty.
Graph Theory . Springer, 2008.[4] K. Culik and D. Wood. A note on some tree similarity measures.
InformationProcessing Letters , 15(1):39–42, 1982.[5] M.M. Deza and M. Laurent.
Geometry of Cuts and Metrics . Springer PublishingCompany, Incorporated, 1st edition, 2009.[6] E. Ehrhart.
Polynômes arithmétiques et méthode des polyèdres en combinatoire .Birkhäuser Verlag, Basel, 1977. International Series of Numerical Mathematics, Vol.35.[7] C. Haase and Tyrrell B. McAllister. Quasi-period collapse and GL n ( Z )-scissors con-gruence in rational polytopes. Contemporary Mathematics , 452(2008):115–122, 2008.168] M. Laurent. Graphic vertices of the metric polytope.
Discrete Mathematics , 151:131–153, 1996.[9] E. Linke. Rational Ehrhart quasi-polynomials.
J. Combin. Theory Ser. A ,118(7):1966–1978, 2011.[10] F. Liu and B. Osserman. Mochizuki’s indigenous bundles and Ehrhart polynomials.
Journal of Algebraic Combinatorics , 23:125–136, 2006.[11] S. Mochizuki.
A Theory of Ordinary p -Adic Curves , volume 32. Publ. RIMS. KyotoUniversity, 1996. 957–1151.[12] D.F. Robinson. Comparison of labeled trees with valency three. Journal of Combi-natorial Theory Ser. B , 11:105–119, 1971.[13] T. Royer. Semi-reflexive polytopes. Available in the ArXiv https://arxiv.org/abs/1712.04381 , 2017.[14] Y. Tsukui. Transformations of cubic graphs.
Journal of the Franklin Institute ,333(B)(4):565–575, 1996.[15] Y. Wakabayashi. Spin networks , Ehrhart quasi-polynomials, and combinatorics ofdormant indigenous bundles. RIMS Preprint 1786, August 2013.[16] D. Zagier. Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula. In