EEIGENPOLYTOPES, SPECTRAL POLYTOPES ANDEDGE-TRANSITIVITY
MARTIN WINTER
Abstract.
Starting from a finite simple graph G , for each eigenvalue θ of itsadjacency matrix one can construct a convex polytope P G ( θ ) , the so called θ -eigenpolytop of G . For some polytopes this technique can be used to recon-struct the polytopes from its edge-graph. Such polytopes (we shall call them spectral ) are still badly understood. We give an overview of the literature foreigenpolytopes and spectral polytopes.We introduce a geometric condition by which to prove that a given poly-tope is spectral (more exactly, θ -spectral). We apply this criterion to the edge-transitive polytopes. We show that every edge-transitive polytope is θ -spectral, is uniquely determined by this graph, and realizes all its symmetries.We give a complete classification of distance-transitive polytopes. Introduction
Eigenpolytopes are a construction in the intersection of combinatorics and geom-etry, using techniques from spectral graph theory. Eigenpolytopes provide a way toassociate several polytopes to a finite simple graph, one for each eigenvalues of itsadjacency matrix. A formal definition can be found in Section 2.2.Eigenpolytopes can be applied from two directions: for the first, one starts froma given graph, computes its eigenpolytopes, and tries to deduce, from the geometryand combinatorics of these polytopes, something about the original graph. For theother direction, one starts with a polytope, asks whether it is an eigenpolytope,and if so, for which graphs, which eigenvalues, and how these relate to the originalpolytope. Eigenpolytopes have several interesting geometric and algebraic proper-ties, and establishing that a family of polytopes consists of eigenpolytopes opensup their study to the techniques of spectral graph theory.For some graphs the connection to their eigenpolytopes is especially strong: it canhappen that a graph is the edge-graph of one of its eigenpolytopes, or equivalently,that a polytope is an eigenpolytope of its edge-graph. Such graphs/polytopes are
Date : September 7, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Eigenpolytopes, spectral polytopes, edge-transitive polytopes, spectralgraph realization. a r X i v : . [ m a t h . M G ] S e p M. WINTER quite special and we shall call them spectral . For example, all regular polytopes arespectral, but there are many others. Their properties are not well-understood.We survey the literature of eigenpolytope and spectral polytopes. We establish atechnique with which to prove that certain polytopes are spectral polytopes and weapply it to edge-transitive polytopes. That are polytopes for which the Euclideansymmetry group
Aut( P ) ⊂ O( R d ) acts transitively on the set of edge F ( P ) . Aswe shall explain, this characterization suffices to proves that an edge-transitivepolytope is uniquely determined by its edge-graph, and also realizes all its combi-natorial symmetries. A complete classification of edge-transitive polytopes is notknown as of yet. However, using results on eigenpolytopes, we are able to give acomplete classification of a sub-class of the edge-transitive polytopes, namely, the distance-transitive polytopes .1.1. Outline of the paper.
Section 2 starts with a motivating example for direct-ing the reader towards the definition of the eigenpolytope as well as the phenomenonof spectral graphs and polytopes. We include a literature overview for eigenpolytopesand spectral polytopes.In Section 3 we give a first rigorous definition for the notion “spectral polytope”via balanced polytopes . The latter is a notion related to the rigidity theory.In Section 4 we introduce the, as of yet, most powerful tool for proving that cer-tain polytopes are spectral.In the final section, Section 5, we apply this result to edge-transitive polytopes.It is a simple corollary of the previous section that these are θ -spectral. We explorethe implications of this finding: edge-transitive polytopes (in dimension d ≥ ) areuniquely determined by the edge-graph and realize all of its symmetries. We discusssub-classes, such as the arc-, half- and distance-transitive polytopes. We close witha complete classification of the latter (based on a result of Godsil).2. Eigenpolytopes and spectral polytopes
A motivating example.
Let G = ( V, E ) be the edge-graph of the cube, withvertex set V = { , ..., } , numbers assigned to the vertices as in the figure below.The spectrum of that graph ( i.e., of its adjacency matrix) is { ( − , ( − , , } .Most often, one denotes the largest eigenvalue by θ , the second-largest by θ , andso on. In spectral graph theory, there exists the general rule of thumb that the mostexciting eigenvalue of a graph is not its largest, but its second-largest eigenvalue θ (which is related to the algebraic connectivity of G ).For the edge-graph of the cube, we have θ = 1 , of multiplicity three . And hereare three linearly independent eigenvectors to θ : IGENPOLYTOPES, SPECTRAL POLYTOPES AND EDGE-TRANSITIVITY 3 u = − − − − , u = − − − − , u = − − − − . We can write these more compactly in a single matrix Φ ∈ R × : Φ = ← v − ← v − ← v − − ← v − ← v − − ← v − − ← v − − − ← v . We now take a look at the rows of that matrix, of which it has exactly eight. Theserows are naturally assigned to the vertices of G (assign i ∈ V to the i -th row of Φ ),and each row can be interpreted as a vector in R .If we place each vertex i ∈ V at the position v i ∈ R given by the i -th row of Φ ,we find that this embedds the graph G exactly as the skeleton of a cube (see thefigure above). In other words: if we compute the convex hull of the v i , we get backthe polyhedron from which we have started. What a coincidence, isn’t it?This example was specifically chosen for its nice numbers, but in fact, the sameworks out as well for many other polytopes, including all the regular polytopes inall dimension. One probably learns to appreciate this magic when suddenly in needfor the vertex coordinates of some not so nice polytope, say, the regular dodeca-hedron or 120-cell. With this technique in the toolbox, these coordinates are justone eigenvector-computation away (we included a short Mathematica script in Ap-pendix A). Note also, that we never specified the dimension of the embedding, butit just so happened, that the second-largest eigenvalue has the right multiplicity.This phenomenon definitely deserves an explanation. On the choice of eigenvectors.
One might object that the chosen eigenvectors u , u and u look suspiciously cherry-picked, and we may not get such a nice result ifwe would have chosen just any eigenvectors. And this is true. For an appropriatechoice of these vectors, we can, instead of a cube, get a cuboid, or a parallelepiped.In fact, we can obtain any linear transformations of the cube. But , we can also get only linear transformations, and nothing else. The reason is the following well knowfact from linear algebra:
Theorem 2.1.
Two matrices Φ , Ψ ∈ R n × d have the same column span, i.e., span Φ =span Ψ , if and only if their rows are related by an invertible linear transformation,i.e., Φ = Ψ T for some T ∈ GL( R d ) . M. WINTER
In our case, the column span is the θ -eigenspace, and the rows are the coordinatesof the v i . We say that any two polytopes constructed in this way are linearlyequivalent .The only notable property of the chosen basis in the example is, that the vectors u , u and u are orthogonal and of the same length. Any other choice of such abasis of the θ -eigenspace ( e.g. an orthonormal basis) would also have given a cube,but reoriented, rescaled and probably with less nice coordinates. For details on howthis choice relates to the orientation, see e.g. [21, Theorem 3.2].2.2. Eigenpolytopes.
We compile our example into a definition.
Definition 2.2.
Start with a graph G = ( V, E ) , an eigenvalue θ ∈ Spec( G ) thereof,as well as an orthonormal basis { u , ..., u d } ⊂ R n of the θ -eigenspace. We define the eigenpolytope matrix Φ ∈ R n × d as the matrix in which the u i are the columns:(2.1) Φ := | | u · · · u d | | = v (cid:62) ... v (cid:62) n . Let v i ∈ R d denote the i -th row of Φ . The polytope P G ( θ ) := conv { v i | i ∈ V } ⊂ R d is called θ -eigenpolytope (or just eigenpolytope ) of G .For later use we define the eigenpolytope map (2.2) φ : V (cid:51) i (cid:55)→ v i ∈ R d that to each vertex i ∈ V assignes the i -th row of the eigenpolytope matrix.Note that the basis { u , ..., u d } ⊂ Eig G ( θ ) in Definition 2.2 is explicitly chosen tobe an orthonormal basis . This is not strictly necessarily, but this choice is convenientfrom a geometric point of view: a different choice for this basis gives the same poly-tope, but with a different orientation rather than, say, transformed by a generallinear transformation. This preserves metric properties and is closer to how poly-topes are usually consider up to rigid motions. We can also reasonably speak of the θ -eigenpolytope, as any two differ only by orientation.With this terminology in place, our observation in the example of Section 2.1 canbe summarized as “the cube is the θ -eigenpolytope of its edge-graph”, or alterna-tively as “the cube-graph is the edge-graph of its θ -eigenpolytope”. Here is a depic-tion of all the eigenpolytopes of the cube-graph, one for each eigenvalue:We observe that the phenomenon from Section 2.1 only happens for θ . In general,the θ -eigenpolytope of a regular graph will always be a single point (which is, whywe rarely care about the largest eigenvalue). Also, whenever a graph is bipartite,the eigenpolytope to the smallest eigenvalue is 1-dimensional, hence a line segment.We are now free to compute the eigenpolytopes of all kinds of graphs, includinggraphs which are not the edge-graph of any polytope (so-called non-polytopal graphs). IGENPOLYTOPES, SPECTRAL POLYTOPES AND EDGE-TRANSITIVITY 5
It is then little surprising that no edge-graph of any of its eigenpolytope gives theoriginal graph again.But even if we start from a polytopal graph, one is not guaranteed to find aneigenpolytope that has the initial graph as its edge-graph ( e.g. the edge-graph of thetriangular prism has no eigenvalue of multiplicity three, hence no eigenpolytope ofdimension three, see also Example 3.4). Equivalently, if one starts with a polytope,it is not guaranteed that this polytope is the eigenpolytope of its edge-graph (oreven combinatorially equivalent to it).
Example 2.3. A neighorly polytope is a polytope whose edge-graph is the completegraph K n . The spectrum of K n is { ( − n − , ( n − } . One checks that the eigen-polytopes are a single point (for θ = n − ) and the regular simplex of dimension n − (for θ = − ).Consequently, no neighborly polytope other than a simplex is combinatoriallyequivalent to an eigenpolytope of its edge-graph.That a graph and its eigenpolytope translate into each other as well as in thecase of the cube in Section 2.1 is a very special phenomenon, to which we shall givea name: a polytope (or graph) for which this happens, will be called spectral . Wecannot formalize this definition right away, as there is some subtlety we have todiscuss first (we give a formal definition in Section 3, see Definition 3.5). Example 2.4.
The image below shows two spectral realizations of the 5-cycle C .The left image shows the realization to the second-largest eigenvalue θ , the rightimage shows the realization to the smallest eigenvalue θ . In both cases, the convexhull (the actual eigenpolytope) is a regular pentagon, whose edge-graph is C again.But we see that only in the case of θ the edges of the graphs get properly mappedinto the edges of the pentagon.While it is true that the 5-cycle C is the edge-graph of its θ -eigenpolytope, theadjacency informations gets scrambled in the process: while, say, vertex 1 and 2 areadjacent in C , their images v and v do not form an edge in the θ -eigenpolytope.We do not want to call this “spectral”, as the adjacency information is not preserved.The same can happen in higher dimensions too, e.g. with G being the edge-graphof the dodecahedron: There was at least one previous attempt to give a name to this phenomenon, namely, in [14],where it was called self-reproducing . Spectral realizations are essentially defined like eigenpolytopes, assinging coordinates v i ∈ R d to each vertex i ∈ V (as in Definition 2.2), but without taking the convex hull. Instead, one drawsthe edges between adjacent vertices. M. WINTER
Observation 2.5.
From studying many examples, there are two interesting obser-vations to be made, both concern θ , none of which is rigorously proven:( i ) It appears as if only θ can give rise to spectral polytopes/graphs. At least,all known examples are θ -spectral (see also Question 6.2). Some consider-ations on nodal domains make this plausible, but no proof is known in thegeneral case (a proof is known in certain special cases, see Theorem 5.7).( ii ) If i ∈ V is a vertex of G , then v i is not necessarily a vertex of every eigen-polytope ( v i might end up in the interior of P G ( θ ) or one of its faces). Andeven if v i , v j ∈ F ( P G ( θ )) are distinct vertices and ij ∈ E is an edge of G ,it is still not necessarily true that conv { v i , v j } is also an edge of the eigen-polytope (as seen in Example 2.4).However, this seems to be no concern in the case θ . It appears as if alledges of G become edges of the θ -eigenpolytope, even if G is not spectral(under mild assumptions on the end vertices of the edge). In other words,the adjacency information of G gets imprinted on the edge-graph of the θ -eigenpolytope, whether G is spectral or not. This is known to be true onlyin the case of distance-regular graphs [10, Theorem 3.3 (b)], but unproven ingeneral (see also Question 6.3)2.3. Litarture.
Eigenpolytope were first introduced by Godsil [9] in 1978. Godsilproved the existence of a group homomorphism
Aut( G ) → Aut( P G ( θ )) , i.e., anycombinatorial symmetry of the graph translates into a Euclidean symmetry of thepolytope. From that, he deduces results about the combinatorial symmetry groupof the original graph.We say more about the group homomorphism: for every θ ∈ Spec( G ) we have Theorem 2.6 ( [9], Theorem 2.2) . If σ ∈ Aut( G ) ⊆ Sym( n ) is a symmetry of G ,and Π σ ∈ Perm( R n ) is the associated permutation matrix, then T σ := Φ (cid:62) Π σ Φ ∈ O( R d ) , (Φ is the eigenpolytope matrix ) is a Euclidean symmetry of the eigenpolytope P G ( θ ) that also permutes the v i as pre-scribed by σ , i.e., T σ ◦ φ = φ ◦ σ , or T σ v i = v σ ( i ) for all i ∈ V . This result is also proven (more generally for spectral graph realizations) in [23,Corollary 2.9].Theorem 2.6 explicitly uses that eigenpolytopes are defined using an orthonormal bases rather than any basis of the eigenspace, to conclude that the symmetries T σ are orthogonal matrices. Also, the statement of Theorem 2.6 is not too satisfyingin general, as it can happen that non-trivial symmetries of G are mapped to theidentity transformation. We not necessarily have Aut( G ) ∼ = Aut( P G ( θ )) . IGENPOLYTOPES, SPECTRAL POLYTOPES AND EDGE-TRANSITIVITY 7
Several authors construct the eigenpolytopes of certain famous graphs or graphfamilies. Powers [18] computed the eigenpolytopes of the
Petersen graph , which hetermed the
Petersen polytopes (one of which will appear as a distance-transitivepolytope in Section 5.4). The same author also investigates eigenpolytopes of gen-eral distance-regular graphs in [19]. In [15], Mohri described the face structure ofthe
Hamming polytopes , the θ -eigenpolytopes of the Hamming graphs. Seeminglyunknown to the author, these polytopes can also by described as the cartesianpowers of regular simplices (also distance transitive, see Section 5.4).There exists a wonderful enumeration of the eigenpolytopes (actually, spectralrealizations) of the edge-graphs of all uniform polyhedra in [3]. Sadly, this write-upwas never published formally. This provides empirical evidence that every uniformpolyhedron has a spectral realization. The same question might then be asked foruniform polytopes in higher dimensions.Rooney [20] used the combinatorial structure of the eigenpolytope (the size oftheir facets) to deduce statements about the size of cocliques in a graph.In [16], the authors investigates how common graph operations translate to op-erations on their eigenpolytopes.Particular attention was given to the eigenpolytopes of distance-regular graphs[8,10,19]. It was shown that in a θ -eigenpolytope of a distance-regular graph G , ev-ery edge of G corresponds to an edge of the eigenpolytope [10]. Consequently, G isa spanning subgraph of the edge-graph of the eigenpolytope. It remains open ifthe same holds for less regular graphs, e.g. i.e., they are spectral in our terminology) was made repeatedly, e.g. in [8] and [14].In the latter, this was shown for all regular polytopes, excluding the exceptional 4-dimensional polytopes, the 24-cell, 120-cell and 600-cell. This gap was filled in [23]via general considerations concerning spectral realizations of arc-transitive graphs.In sum, all regular polytopes are known to be θ -spectral.The next major result for spectral polytopes was obtained by Godsil in [10],where he was able to classify all θ -spectral distance-regular graphs (see also Sec-tion 5.4): Theorem 2.7 ([10], Theorem 4.3) . Let G be distance-regular. If G is θ -spectral, then G is one of the following: ( i ) a cycle graph C n , n ≥ , ( ii ) the edge-graph of the dodecahedron, ( iii ) the edge-graph of the icosahedron, ( iv ) the complement of a disjoint union of edges, ( v ) a Johnson graph J ( n, k ) , ( vi ) a Hamming graph H ( d, q ) , ( vii ) a halved n -cube / Q n , ( viii ) the Schläfli graph, or ( ix ) the Gosset graph. A second look at this list reveals a remarkable “coincidence”: while the genericdistance-regular graph has few or no symmetries, all the graphs in this list are highlysymmetric, in fact, distance-transitive (a definition will be given in Section 5.4).It is a widely open question whether being spectral is a property solely reservedfor highly symmetric graphs and polytopes (see also Question 6.4). There is only
M. WINTER a single known spectral polytope that is not vertex-transitive (see also Remark 5.3and Question 6.5). 3.
Balanced and spectral polytopes
In this section we give a second approach to spectral polytopes that circumventsthe mentioned subtleties.For the rest of the paper, let P ⊂ R d denote a full-dimensional polytope in dimen-sion d ≥ with vertices v , ..., v n ∈ F ( P ) . We disinguish the skeleton of P , whichis the graph with vertex set F ( P ) and edge set F ( P ) , from the edge-graph G P =( V, E ) of P , which is isomorphic to the skeleton, but has vertex set V = { , ..., n } .The isomorphism will be denoted(3.1) ψ : V (cid:51) i (cid:55)→ v i ∈ F ( P ) , and we call it the skeleton map .3.1. Balanced polytopes.Definition 3.1.
The polytope P is called θ -balanced (or just balanced ) for some realnumber θ ∈ R , if(3.2) (cid:88) j ∈ N ( i ) v j = θv i , for all i ∈ V , where N ( i ) := { j ∈ V | ij ∈ E } denotes the neighborhood of a vertex i ∈ V .One way to interpret the balancing condition (3.2) is as a kind of self-stress con-dition on the skeleton of P (the term “balanced” is motivated from this). For eachedge ij ∈ E , the vector v j − v i is parallel to the edge conv { v i , v j } . If P is θ -balanced,at each vertex i ∈ V we have the equation (cid:88) j ∈ N ( i ) ( v j − v i ) = (cid:88) j ∈ N ( i ) v j − deg( i ) v i = (cid:0) θ − deg( i ) (cid:1) v i . This equation can be interpreted as two forces that cancel each other out: on theleft, a contracting force along each edge (proportion only to the length of that edge),and on the right, a force repelling each vertex away from the origin (proportionalto the distance of that vertex from the origin, and proportional to θ − deg( i ) ).A second interpretation of (3.2) is via spectral graph theory. Define the matrix(3.3) Ψ := v (cid:62) ... v (cid:62) n in which the v i are the rows. This matrix will be called the arrangement matrix of P . Note that the skeleton map ψ assignes i ∈ V to the i -th row of Ψ . Since we usethat P ⊂ R d is full-dimensional, we have rank Ψ = d . Observation 3.2.
Suppose that P is θ -balanced. The defining equation (3.2) canbe equivalently written as the matrix equation A Ψ = θ Ψ . In this form, it is apparentthat θ is an eigenvalue of the adjacency matrix A , and the columns of Ψ are θ -eigenvectors, or span Ψ ⊆ Eig G P ( θ ) .We have seen that for a balanced polytope, the columns of Ψ must be eigenvec-tors. But they are not necessarily a complete set of θ -eigenvectors, i.e., they notnecessarily span the whole eigenspace. IGENPOLYTOPES, SPECTRAL POLYTOPES AND EDGE-TRANSITIVITY 9
Example 3.3.
Every centered neighborly polytope P is balanced, but except if it isa simplex, it is not spectral (the latter was shown in Example 2.3). Centered meansthat (cid:88) i ∈ V v i = 0 . Since P is neighborly, we have G P = K n and N ( i ) = V \{ i } for all i ∈ V . Therefore (cid:88) j ∈ N ( i ) v j = (cid:88) j ∈ V v j − v i = − v i , for all i ∈ V .
And indeed, K n has spectrum { ( − n − , ( n − } . So P is ( − -balanced.The last example shows that every neighborly polytopes can be made balancedby merely translating it. More generally, many polytopes have a realization (of theircombinatorial type) that is balanced. But other polytopes do not: Example 3.4.
Let P ⊂ R be a triangular prism.The spectrum of the edge-graph of P is { ( − , , , } . Note that there is noeigenvalue of multiplicity greater than two. In particular, we cannot choose threelinearly independent eigenvectors to a common eigenvalue. But if P were balanced,then Observation 3.2 tells us that the columns of the arrangement matrix Ψ wouldbe three eigenvectors to the same eigenvalue (linearly independent, since rank Ψ =3 ), which is not possible. And so, no realization of P can be balanced.3.2. Spectral graphs and polytopes.
In the extreme case, when the columns of Ψ span the whole eigenspace, we can finally give a compact definition of what wewant to consider as spectral : Definition 3.5. ( i ) A polytope P is called θ -spectral (or just spectral ), if its arrangement matrix Ψ satisfies span Ψ = Eig G P ( θ ) .( ii ) A graph is said to be θ -spectral (or just spectral ) if it is (isomorphic to) theedge-graph of a θ -spectral polytope.This definition is now perfectly compatible with our initial motivation for the term“spectral” in Section 2.2. Lemma 3.6. ( i ) If a polytope P is θ -spectral, then P is linearly equivalent to the θ -eigenpoly-tope of its edge-graph (see also Proposition 3.7). ( ii ) If a graph G is θ -spectral, then G is (isomorphic to) the edge-graph of its θ -eigenpolytope (see also Proposition 3.8). In both cases, the converse is not true. This is intentional, to avoid the problemsmentioned in Example 2.4. Both statement will be proven below by formulating amore technical condition that is then actually equivalent to being spectral.
Proposition 3.7.
A polytope P is θ -spectral if and only if it is linearly equivalentto the θ -eigenpolytope of its edge-graph via some linear map T ∈ GL( R d ) for whichthe following diagram commutes: (3.4) P P G P ( θ ) G PTψ φ where φ and ψ denote the eigenpolytope map and skeleton map respectively.Proof. By definition, the θ -eigenpolytope of G P satisfies span Φ = Eig G P ( θ ) , where Φ is the corresponding eigenpolytope matrix.Now, by definition, P is θ -spectral if and only if span Ψ = Eig G P ( θ ) , where Ψ isits arrangement matrix. But by Theorem 2.1, Φ and Ψ have the same span if andonly of their rows are related by some invertible linear map T ∈ GL( R d ) , that is, Ψ T = Φ , or T ◦ ψ = φ . The latter expresses exactly that (3.4) commutes. (cid:3) This also proves Lemma 3.6 ( i ) . Proposition 3.8.
A graph G is θ -spectral if and only if the eigenpolytope map φ : V ( G ) → R d provides an isomorphism between G and the skeleton of its θ -eigenpoly-tope P G ( θ ) .Proof. Suppose first that G is θ -spectral. Then there is a θ -spectral polytope Q withedge-graph G Q = G and skeleton map ψ : V ( G Q ) → F ( Q ) . By Lemma 3.6 ( i ) , Q islinearly equivalent to P G ( θ ) via some linear map T ∈ GL( R d ) . By Proposition 3.7,the eigenpolytope map satisfies φ = T ◦ ψ . Since T induces an isomorphism betweenthe skeleta of Q and P G ( θ ) , and ψ is an isomorphism between G and the skeletonof Q , we find that φ must be an isomorphism between G and the skeleton of P G ( θ ) .This shows one direction.For the converse, suppose that φ is an isomorphism. Set P := P G ( θ ) and let G P be its edge-graph with skeleton map ψ : V ( G P ) → F ( P ) . Then σ := ψ − ◦ φ is agraph isomorphism between G and G P . So, since G ∼ = G P , each eigenpolytope of G is also an eigenpolytope of G P . We can therefore choose P G P ( θ ) = P G ( θ ) , withcorresponding eigenpolytope map φ (cid:48) := σ − ◦ φ . In sum, the outer square in thefollowing diagram commutes: G G P P := P G ( θ ) P G P ( θ ) σφ φ (cid:48) ψ Id Also, by construction of σ , the upper triangle commutes. In conclusion, the lowertriangle must commute as well, which is exactly (3.4) with T = Id . This proves that P is θ -spectral via Proposition 3.7. Since G is isomorphic to G P , G is θ -spectral. (cid:3) This also proves Lemma 3.6 ( ii ) .It is also possible to give a definition of spectral graphs purely in terms of graphtheory, without any explicit reference to polytopes: Lemma 3.9.
A graph G is θ -spectral if and only if it satisfies both of the following: ( i ) for each vertex i ∈ V exists a θ -eigenvector u = ( u , ..., u n ) ∈ Eig G ( θ ) whosesingle largest component is u i , or equivalently, Argmax k ∈ V u k = { i } . ( ii ) any two vertices i, j ∈ V form an edge ij ∈ E in G if and only if there is a θ -eigenvector u = ( u , ..., u n ) ∈ Eig G ( θ ) whose only two largest componentsare u i and u j , or equivalently, Argmax k ∈ V u k = { i, j } . IGENPOLYTOPES, SPECTRAL POLYTOPES AND EDGE-TRANSITIVITY 11
This characterization of spectral graphs can be interpreted as follows: a spectralgraph can be reconstructed from knowing a single eigenspace, rather than, say, alleigenspaces and their associated eigenvalues.
Proof of Lemma 3.9.
Let P G ( θ ) ⊂ R d be the θ -eigenpolytope of G with eigenpoly-tope matrix Φ and eigenpolytope map φ : V (cid:51) i (cid:55)→ v i ∈ R d .Since span Φ = Eig G ( θ ) , the eigenvectors u = ( u , ..., u n ) ∈ Eig G ( θ ) are exactlythe vectors that can be written as u = Φ x for some x ∈ R d . If then e k ∈ R n denotesthe k -th standard basis vector, we have u k = (cid:104) u, e k (cid:105) = (cid:104) Φ x, e k (cid:105) = (cid:104) x, Φ (cid:62) e k (cid:105) = (cid:104) x, v k (cid:105) . Therefore, there is a θ -eigenvector u = ( u , ..., u n ) ∈ Eig G ( θ ) with Argmax k ∈ V u k = { i , ..., i m } if and only if there is a vector x ∈ R d with Argmax k ∈ V (cid:104) x, v k (cid:105) = { i , ..., i m } . But this last line is exactly what it means for conv { v i , ..., v i m } to be a face of P G ( θ )= conv { v , ..., v n } (and x is a normal vector of that face).In this light, we can interpret ( i ) as stating that v , ..., v n form n distinct verticesof P G ( θ ) , and ( ii ) as stating that conv { v i , v j } is an edge of P G ( θ ) if and only if ij ∈ E . And this means exactly that φ is a graph isomorphism between G and theskeleton of P G ( θ ) . By Proposition 3.8, this is equivalent to G being θ -spectral. (cid:3) In practice, to reconstruct a spectral graph from an eigenspace, the steps couldbe the following: given a subspace U ⊆ R n (the claimed eigenspace), then( i ) choose any basis u , ..., u d ∈ R n of U ,( ii ) build the matrix Φ = ( u , ..., u d ) ∈ R n × d in which the u i are the columns,( iii ) define v i as the i -th row of Φ ,( iv ) define P := conv { v , ..., v n } ⊂ R d as the convex hull of the v i ,( v ) the reconstructed graph G = G P is then the edge-graph of P .3.3. Properties of spectral polytopes.
We discuss two properties of spectralpolytopes that make them especially interesting in polytope theory.
Reconstruction from the edge-graph.
The edge-graph of a general polytope carrieslittle information about that polytope i.e., given only its edge-graph, we can oftennot reconstruct the polytope from this (up to combinatorial equivalence). Often,one cannot even deduce the dimension of the polytope from its edge-graph. Re-construction might be possible in certain special cases, as e.g. for 3-dimensionalpolyhedra, simple polytopes or zonotopes. The spectral polytopes provide anothersuch class.
Theorem 3.10. A θ k -spectral polytope is uniquely determined by its edge-graph upto invertible linear transformations. The proof is simple: every θ k -spectral polytope is linearly equivalent to the θ k -eigenpolytope of its edge-graph (by Lemma 3.6 ( i ) ). Our definition of the θ k -eigen-polytope already suggests an explicit procedure to construct it (a script for this isincluded in Appendix A). This property of spectral polytopes appears more excitingwhen applied to graph classes that are not obviously spectral (see Section 5). Realizing symmetries of the edge-graph.
Every Euclidean symmetry of a polytopeinduces a combinatorial symmetry on its edge-graph. The converse is far fromtrue. Think, for example, about a rectangle that is not a square. Even worse, itcan happen that a polytope does not even have a realization that realizes all thesymmetries of its edge-graph ( e.g. the polytope constructed in [4]).We have previously discussed (in Theorem 2.6) the existence of a homomorphism
Aut( G ) → Aut( P G ( θ )) between the symmetries of a graph G and the symmetriesof its eigenpolytopes. There are two caveats:( i ) this is not necessarily an isomorphism, and( ii ) it says nothing about the symmetries of the edge-graph of P G ( θ ) , as thisone needs not to be isomorphic to G Still, it suffices to makes statement of the following form: if G is vertex-transitive,then so are all its eigenpolytopes. This might not work with other transitivities, asfor example edge-transitivity.This is no concern for spectral graphs/polytopes: Theorem 3.11. ( i ) If G is θ -spectral, then P G ( θ ) realizes all its symmetries, which includes Aut( G ) ∼ = Aut( P G ( θ )) via the map σ (cid:55)→ T σ given in Theorem 2.6, as wells as that T σ permutes thevertices and edges of P G ( θ ) exactly as σ permutes the vertices and edges ofthe graph G . ( ii ) If P is θ -spectral, then P has a realization that realizes all the symmetries ofits edge-graph, namely, the θ -eigenpolytope of its edge-graph. This is mostly straight forward, with large parts already addressed in Theorem 2.6.The major difference is that for spectral graphs G the eigenpolytope has exactly thedistinct vertices v , ..., v n ∈ R d . The statement from Theorem 2.6 that T σ permutesthe v i as prescribed by σ , then becomes, that T σ permutes the vertices as prescribedby σ , and hence also the edges. Also, since the v i are distinct, no non-trivial sym-metry σ can result in trivial T σ , making σ (cid:55)→ T σ into a group isomorphism .For part ( ii ) merely recall that the eigenpolytope P G P ( θ ) is indeed a realizationof P by Lemma 3.6 ( i ) .The major consequence of this is, that for spectral graphs/polytopes also morecomplicates types of symmetries translate between a polytope and its graph, as e.g. edge-transitivity (see also Section 5).4. The Theorem of Izmestiev
We introduce our, as of yet, most powerful tool for proving that certain polytopesare θ -spectral. For this, we make use of a more general theorem by Izmestiev [13],first proven in the context of the Colin de Verdière graph invariant. The proofof this theorem requires techniques from convex geometry, most notably, mixedvolumes, which we not address here. We need to introduce some terminology.As before, let P ⊂ R d denote a full-dimensional polytope of dimension d ≥ ,with edge-graph G P = ( V, E ) , V = { , ..., n } and vertices v i ∈ F ( P ) , i ∈ V . Recall,that the polar dual of P is the polytope P ◦ := { x ∈ R d | (cid:104) x, v i (cid:105) ≤ for all i ∈ V } . IGENPOLYTOPES, SPECTRAL POLYTOPES AND EDGE-TRANSITIVITY 13
We can replace the -s in this definition by variables c = ( c , ..., c n ) , to obtain P ◦ ( c ) := { x ∈ R d | (cid:104) x, v i (cid:105) ≤ c i for all i ∈ V } . The usual polar dual is then P ◦ = P ◦ (1 , ..., . Figure 1.
Visualization of P ◦ ( c ) for different values of c ∈ R n . In the following, vol( · ) denotes the volume of convex sets in R d ( w.r.t. the usualLebesgue measure). Note that the function vol( P ◦ ( c )) is differentiable in c , and sowe can compute partial derivatives w.r.t. the components of c . Theorem 4.1 (Izmestiev [13], Theorem 2.4) . Define a matrix X ∈ R n × n with compo-nents X ij := − ∂ vol( P ◦ ( c )) ∂c i ∂c j (cid:12)(cid:12)(cid:12) c =(1 ,..., . The matrix X has the following properties: ( i ) X ij < whenever ij ∈ E ( G P ) , ( ii ) X ij = 0 whenever ij (cid:54)∈ E ( G P ) , ( iii ) X Ψ = 0 (where Ψ is the arrangement matrix of P ), ( iv ) X has a unique negative eigenvalue, and this eigenvalue is simple, ( v ) dim ker X = d . One can view the matrix X as some kind of adjacency matrix of a vertex- andedge-weighted version of G P . Part ( iii ) states that v satisfies a weighted form of thebalancing condition (3.2) with eigenvalue zero. Since rank Ψ = d , part ( v ) statesthat span Ψ is already the whole 0-eigenspace. And part ( iv ) states that zero is thesecond smallest eigenvalue of X . Theorem 4.2.
Let X ∈ R n × n be the matrix defined in Theorem 4.1. If we have ( i ) X ii is independent of i ∈ V ( G P ) , and ( ii ) X ij is independent of ij ∈ E ( G P ) ,then P is θ -spectral.Proof. By assumption there are α, β ∈ R , β > , so that X ii = α for all vertices i ∈ V ( G P ) , and X ij = β < for all edges ij ∈ E ( G P ) (we have β < by Theorem 4.1 ( i ) ). We can write this as X = α Id + βA = ⇒ ( ∗ ) A = αβ Id + 1 β X, where A is the adjacency matrix of G P . By Theorem 4.1 ( iv ) and ( v ) , the matrix X has second smallest eigenvalue zero of multiplicity d . By Theorem 4.1 ( iii ) , thecolumns of M are the corresponding eigenvectors. Since rank Ψ = d we find thatthese are all the eigenvectors and span Ψ is the 0-eigenspace of X . By ( ∗ ) the eigenvalues of A are the eigenvalues of X , but scaled by /β andshifted by α/β . Since /β < , the second- smallest eigenvalue of X gets mappedonto the second- largest eigenvalue of A . Therefore, A (and also G P ) has second-largest eigenvalue θ = α/β of multiplicity d , and span Ψ is the correspondingeigenspace. By definition, P is then the θ -eigenpolytope of G P and is therefore θ -spectral. (cid:3) It is unclear whether Theorem 4.2 already characterizes θ -spectral polytopes,or even spectral polytopes in general (see also Question 6.1).5. Edge-transitive polytopes
We apply Theorem 4.2 to edge-transitive polytopes, that is, to polytopes for whichthe Euclidean symmetry group
Aut( P ) ⊂ O( R d ) acts transitively on the edge set F ( P ) . No classification of edge-transitive polytopes is known. Some edge-transitivepolytopes are listed in Section 5.2.Despite the name of this section, we are actually going to address polytopes thatare simultaneously vertex- and edge-transitive. This is not a huge deviation fromthe title: as shown in [22], edge-transitive polytopes in dimension d ≥ are alwaysalso vertex-transitive, and the exceptions in lower dimensions are few (a continuousfamily of n -gons for each n ≥ , and two exceptional polyhedra).Theorem 4.2 can be directly applied to simultaneously vertex- and edge-transitivepolytopes, and so we have Corollary 5.1.
A simultaneously vertex- and edge-transitive polytope is θ -spectral. We collect all the notable consequences in the following theorem:
Theorem 5.2. If P ⊂ R d is simultaneously vertex- and edge-transitive, then ( i ) Aut( P ) ⊂ R d is irreducible as a matrix group. ( ii ) P is uniquely determined by its edge-graph up to scale and orientation. ( iii ) P realizes all the symmetries of its edge-graph. ( iv ) if P has edge length (cid:96) and circumradius r , then (5.1) (cid:96)r = (cid:115) λ deg( G P ) = (cid:115) (cid:16) − θ deg( G P ) (cid:17) , where deg( G P ) is the vertex degree of G P , and λ = deg( G P ) − θ denotesits second smallest Laplacian eigenvalue. ( v ) if α is the dihedral angle of the polar dual P ◦ , then (5.2) cos( α ) = − θ deg( G P ) . Proof.
The complete proof of ( i ) and ( ii ) has to be postponed until Section 5.1 (seeTheorem 5.4). Concerning ( ii ) , from Corollary 5.1 and Theorem 3.10 already followsthat P is determined by its edge-graph up to invertible linear transformations , butnot necessarily only up to scale and orientation.Part ( iii ) follows from Theorem 3.11. Part ( iv ) and ( v ) were proven (in a moregeneral setting) in [23, Proposition 4.3]. This applies literally to ( iv ) . For ( v ) , notethe following: if σ i ∈ F d − ( P ◦ ) is the facet of the polar dual P ◦ that corresponds This shows that P is perfect , i.e., is the unique maximally symmetric realization of its com-binatorial type. See [7] for an introduction to perfect polytopes. IGENPOLYTOPES, SPECTRAL POLYTOPES AND EDGE-TRANSITIVITY 15 to the vertex v i ∈ F ( P ) , then the dihedral angle between σ i and σ j is π − (cid:93) ( v i , v j ) .The latter expression was proven in [23] to agree with (5.2). (cid:3) It is worth emphasizing that large parts of Theorem 5.2 do not apply to polytopesof a weaker symmetry, as e.g. vertex-transitive polytopes. Prisms are counterexam-ples to both ( i ) and ( ii ) . There are vertex-transitive neighborly polytopes (otherthan simplices) and they are counterexamples to ( ii ) and ( iii ) . Remark 5.3.
There are two edge-transitive polyhedra that are not vertex-transitive:the rhombic dodecahedron and the rhombic triacontahedron (see also Figure 2). Onlythe former is θ -spectral, and the latter is not spectral for any eigenvalue (this wasalready mentioned in [14]). Since the rhombic dodecahedron is not vertex-transitive,nothing of this follows from Corollary 5.1. However, this polytope satisfies the con-ditions of Theorem 4.2, which seems purely accidental. It is the only known spectralpolytope that is not vertex-transitive.5.1. Rigidity and irreducibility.
The goal of this section is to prove the missingpart of Theorem 5.2:
Theorem 5.4. If P ⊂ R d is simultaneously vertex- and edge-transitive, then ( i ) Aut( P ) ⊂ O( R d ) is irreducible as a matrix group, and ( ii ) P is determined by its edge-graph up to scale and orientation. To prove Theorem 5.4, we make use of
Cauchy’s rigidity theorem for polyhedra(with its beautiful proof listed in [2, Section 12]). It states that every polyhedron isuniquely determined by its combinatorial type and the shape of its faces. This wasgeneralized by Alexandrov to general dimensions d ≥ (proven e.g. in [17, Theorem27.2]): Theorem 5.5 (Alexandrov) . Let P , P ⊂ R d , d ≥ be two polytopes, so that ( i ) P and P are combinatorially equivalent via a face lattice isomorphism φ : F ( P ) → F ( P ) , and ( ii ) each facet σ ∈ F d − ( P ) is congruent to the facet φ ( σ ) ∈ F d − ( P ) .Then P and P are congruent, i.e., are the same up to orientation. Proposition 5.6.
Let P , P ⊂ R d be two combinatorially equivalent polytopes, eachof which has ( i ) all vertices on a common sphere (i.e., is inscribed), and ( ii ) all edges of the same length (cid:96) i .Then P and P are the same up to scale and orientation.Proof. W.l.o.g. assume that P and P have the same circumradius, otherwise re-scale P . It then suffices to show that P and P are the same up to orientation.We proceed with induction by the dimension d . The induction base is given by d = 2 , which is trivial, since any two inscribed polygons with constant edge lengthare regular and thus completely determined (up to scale and orientation) by theirnumber of vertices.Suppose now that P and P are combinatorially equivalent polytopes of dimen-sion d ≥ that satisfy ( i ) and ( ii ) . Let φ be the face lattice isomorphism betweenthem. Let σ ∈ F d − ( P ) be a facet of P , and φ ( σ ) the corresponding facet in This proof was proposed by the user
Fedor Petrov on MathOverflow [1]. P . In particular, σ and φ ( σ ) are combinatorially equivalent. Furthermore, both σ and φ ( σ ) are of dimension d − and satisfy ( i ) and ( ii ) . This is obvious for ( ii ) , and for ( i ) recall that facets of inscribed polytopes are also inscribed. Byinduction hypothesis, σ and φ ( σ ) are then congruent. Since this holds for all facets σ ∈ F d − ( P ) , Theorem 5.5 tells us that P and P are congruent, that is, the sameup to orientation. (cid:3) We can now prove the main theorem of this section:
Proof of Theorem 5.4.
By Theorem 5.2 the combinatorial type of P is determinedby its edge-graph. By vertex-transitivity, all vertices are on a sphere. By edge-transitivity, all edges are of the same length. We can then apply Proposition 5.6 toobtain that P is unique up to scale and orientation. This proves ( ii ) .Suppose now, that Aut( P ) is not irreducible, but that R d decomposes as R d = W ⊕ W into non-trivial orthogonal Aut( P ) -invariant subspaces. Let T α ∈ GL( R d ) be the linear map that acts as identity on W , but as α Id on W for some α > .Then T α P is a non-orthogonal linear transformation of P (in particular, combina-torially equivalent), on which Aut( P ) still acts vertex- and edge-transitively. By ( ii ) , this cannot be. Hence Aut( P ) must be irreducible, which proves ( i ) . (cid:3) A word on classification.
Despite the simple appearance of the definitionof an edge-transitive polytope, no classification was obtained so far.There exists a classification of the 3-dimension edge-transitive polyhedra: besidesthe Platonic solids, these are the ones shown in Figure 2 (nine in total).
Figure 2.
From left to right, these are: the cuboctahedron, the icosido-decahedron, the rhombic dodecahedron, and the rhombic triacontahe-dron.
There are many known edge-transitive polytopes in dimension d ≥ (so we arenot talking about a class as restricted as the regular polytopes). There are 15known edge-transitive 4-polytopes (and an infinite family of duoprisms ), but al-ready here, no classification is known. It is known that the number of irreducible edge-transitive polytopes grows at least linearly with the number of dimensions. Forexample, there are (cid:98) d/ (cid:99) hyper-simplices in dimension d . These are edge-transitive(even distance-transitive, see Section 5.4).It is the hope of the author, that the classification of the edge-transitive polytopescan be obtained using their spectral properties. Their classification can now be The ( n, m ) -duoprism is the cartesian product of a regular n -gon and a regular m -gon. Thoseare edge-transitive if and only of n = m . Technically, the 4-cube is the (4 , -duoprism but isusually not counted as such, because of its exceptionally large symmetry group. Being not the cartesian product of lower dimensional edge-transitive polytopes.
IGENPOLYTOPES, SPECTRAL POLYTOPES AND EDGE-TRANSITIVITY 17 stated purely as a problem in spectral graph theory: the classification of the edge-transitive polytopes (in dimension d ≥ ) is equivalent to the classification of θ -spectral edge-transitive graphs, and since Lemma 3.9, we have a completely graphtheoretic characterization of spectral graphs. Theorem 5.7.
Let G be an edge-transitive graph. If G is θ k -spectral, then ( i ) k = 2 , and ( ii ) if G is not vertex-transitive, then G is the edge-graph of the rhombic dodec-ahedron (see Figure 2).Proof. We first prove ( ii ) . As shown in [22] all edge-transitive polytopes in dimen-sion d ≥ are vertex-transitive. If G is edge-transitive, not vertex-transitive and θ k -spectral, then its θ k -eigenpolytope is also edge-transitive but not vertex-transitive,hence of dimension d ≤ . One checks that the 2-dimensional spectral polytopes areregular polygons, hence vertex-transitive. The remaining polytopes are polyhedra,and we mentioned in Remark 5.3 that among these, only the rhombic dodecahedronis spectral, in fact θ -spectral. This proves ( ii ) .Equivalently, if G is vertex- and edge-transitive, then so is its eigenpolytope. ByCorollary 5.1 this is a θ -eigenpolytope. Together with part ( ii ) , we find k = 2 in allcases, which proves ( i ) . (cid:3) Arc- and half-transitive polytopes.
In a graph or polytope, an arc is anincident vertex-edge-pair. A graph or polytope is called arc-transitive if its sym-metry group acts transitively on the arcs. Being arc-transitive implies both, beingvertex-transitive, and being edge-transitive. In addition to that, in an arc-transitivegraph, every edge can be mapped, not only onto every other edge, but also ontoitself with flipped orientation.There exist graphs that are simultaneously vertex- and edge-transitive, but notarc-transitive. Those are called half-transitive graphs, and are comparatively rare.The smallest one has vertices and is known as the Holt graph (see [5, 12]).For polytopes on the other hand, it is unknown whether there eixsts a distinctionbeing arc-transitive and being simultaneously vertex- and edge-transitive. No half-transitive polytope is known. Because of Theorem 5.2 ( i ) , we know that the edge-graph of a half-transitive polytope must itself be half-transitive. Since such graphsare rare, the existence of half-transitive polytopes seems unlikely. Example 5.8.
The Holt graph is not the edge-graph of a half-transitive polytope:the Holt graph is of degree four, and its second-largest eigenvalue is of multiplicitysix, giving rise to a 6-dimensional θ -eigenpolytope. But a 6-dimensional polytopemust have an edge-graph of degree at least six, and so the Holt graph is not spectral.The lack of examples of half-transitive polytopes means that all known edge-transitive polytopes in dimension d ≥ are in fact arc-transitive. Likewise, a clas-sification of arc-transitive polytopes is not known.5.4. Distance-transitive polytopes.
Our previous results about edge-transitivepolytopes already allow for a complete classification of a particular subclass, namely,the distance-transitive polytopes , thereby also providing a list of examples of edge-transitive polytopes in higher dimensions.The distance-transitive symmetry is usually only considered for graphs, and thedistance-transitive graphs form a subclass of the distance-regular graphs. The usualreference for these is the classic monograph by Brouwer, Cohen and Neumaier [6].
For any two vertices i, j ∈ V of a graph G , let dist( i, j ) denote the graph-theoretic distance between those vertices, that is, the length of the shortest path connectingthem. The diameter diam( G ) of G is the largest distance between any two verticesin G . Definition 5.9.
A graph is called distance-transitive if Aut( G ) acts transitivelyon each of the sets D δ := { ( i, j ) ∈ V × V | dist( i, j ) = δ } , for all δ ∈ { , ..., diam( G ) } . Analogously, a polytope P ⊂ R d is said to be distance-transitive , if its Euclideansymmetry group Aut( P ) acts transitively on each of the sets D δ := { ( v i , v j ) ∈ F ( P ) × F ( P ) | dist( i, j ) = δ } , for all δ ∈ { , ..., diam( G P ) } . Note that the distance between the vertices is still measured along the edge-graphrather than via the Euclidean distance.Being arc-transitive is equivalent to being transitive on the set D . Hence,distance-transitivity implies arc-transitivity, thus edge-transitivity.By our considerations in the previous sections, we know that the classificationof distance-transitive polytopes is equivalent to the classification of the θ -spectraldistance-transitive graphs. Those where classified by Godsil (see Theorem 2.7).In the following theorem we translated each such θ -spectral distance-transitivegraph into its respective eigenpolytope. This gives a complete classification of thedistance-transitive polytopes. Theorem 5.10. If P ⊂ R d is distance-transitive, then it is one of the following: ( i ) a regular polygon ( d = 2) , ( ii ) the regular dodecahedron ( d = 3) , ( iii ) the regular icosahedron ( d = 3) , ( iv ) a cross-polytopes, that is, conv {± e , ..., ± e d } where { e , ..., e d } ⊂ R d is thestandard basis of R d , ( v ) a hyper-simplex ∆( d, k ) , that is, the convex hull of all vectors v ∈ { , } d +1 with exactly k ( vi ) a cartesian power of a regular simplex (also known as the Hamming poly-topes; this includes regular simplices and hypercubes), ( vii ) a demi-cube, that is, the convex hull of all vectors v ∈ {− , } d with an evennumber of 1-entries, ( viii ) the -polytope, also called Gosset-polytope ( d = 6) , ( ix ) the -polytope, also called Schläfli-polytope ( d = 7) .The ordering of the polytopes in this list agrees with the ordering of graphs in the listin Theorem 2.7. The latter two polytopes where first constructed by Gosset in [11]. We observe that the list in Theorem 5.10 contains many polytopes that are notregular, and contains all regular polytopes excluding the 4-dimensional exceptions,the 24-cell, 120-cell and 600-cell. The distance-transitive polytopes thus form adistinct class of remarkably symmetric polytopes which is not immediately relatedto the class of regular polytopes.Another noteworthy observation is that all the distance-transitive polytopes are
Wythoffian polytopes , that is, they are orbit polytopes of finite reflection groups.Figure 3 shows the Coxeter-Dynkin diagrams of these polytopes.
IGENPOLYTOPES, SPECTRAL POLYTOPES AND EDGE-TRANSITIVITY 19
Figure 3.
Coxeter-Dynkin diagrams of distance-transitive polytopes. Conclusion and open questions
In this paper we have studied eigenpolytopes and spectral polytopes . The formerare polytopes constructed from a graph and one of its eigenvalues. A polytope isspectral if it is the eigenpolytopes of its edge-graph. These are of interest becausespectral graph theory then ensures a strong interplay between the combinatorialproperties of the edge-graph and the geometric properties of the polytope.The study of eigenpolytopes and spectral polytopes has left us with many openquestions. Most notably, how to detect spectral polytopes purely from their geome-try. We introduced a tool (Theorem 4.2), which was sufficient to proof that (most)edge-transitive polytopes are spectral. We do not know how much more general itcan be applied.
Question 6.1.
Does Theorem 4.2 already characterize θ -spectral polytopes (oreven spectral polytopes in general)?If the answer is affirmative, this would provide a geometric characterization ofpolytopes that are otherwise defined purely in terms of spectral graph theory. Theresult of Izmestiev suggests that polytopes with sufficiently regular geometry are θ -spectral: the entry of the matrix X in Theorem 4.1 at index ij ∈ E can be expressedas X ij = vol( σ i ∩ σ j ) (cid:107) v i (cid:107)(cid:107) v j (cid:107) sin (cid:93) ( v i , v j ) , where σ i and σ j are the facets of the polar dual P ◦ that correspond to the vertices v i , v j ∈ F ( P ) . Because of this formula, it might be actually easier to classify thepolar duals of θ -spectral polytopes.An affirmative answer to Question 6.1 would also mean a negative answer to thefollowing: Question 6.2.
Is there a θ k -spectral polytope/graph for some k (cid:54) = 2 ?The answer is known to be negative for edge-transitive polytopes/graphs (seeTheorem 5.7), but unknown in general.The second-largest eigenvalue θ is special for other reasons too. Even if a graphis not θ -spectral, it seems to still imprint its adjacency information onto the edge-graph of its θ -eigenpolytope. Question 6.3.
Given an edge ij ∈ E of G , if v i and v j (as defined in Definition 2.2)are distinct vertices of the θ -eigenpolytope P G ( θ ) , is then also conv { v i , v j } an edgeof P G ( θ ) ?This was proven for distance-regular graphs in [10], and is not necessarily truefor eigenvalues other than θ .All known spectral polytopes are exceptionally symmetric. It is unclear whetherthis is true in general. Question 6.4.
Are there spectral polytopes with trivial symmetry group?An example for Question 6.4 must be asymmetric, yet with a reasonably largeeigenspaces. Such graphs exist among the distance-regular graphs, but all spectraldistance-regular graphs were determined in [10] (see also Theorem 2.7) and turnedout to be distance-transitive, i.e., highly symmetric.A clear connection between being spectral and being symmetric is missing. Toemphasize our ignorance, we ask the following:
Question 6.5.
Can we find more spectral polytopes that are not vertex-transitive ?What characterizes them?The single known spectral polytope that is not vertex-transitive is the rhombicdodecahedron (see Figure 2). The fact that it is spectral appears purely accidental,as there seems to be no reason for it to be spectral, except that we can explicitlycheck that it is. For comparison, the highly related rhombic triacontahedron is notspectral.On the other hand, vertex-transitive spectral polytopes might be quite common.
Question 6.6.
Let P ⊂ R d be a polytope with the following properties:( i ) P is vertex-transitive,( ii ) P realizes all the symmetries of its edge-graph, and( iii ) Aut( P ) is irreducible.Is P (combinatorially equivalent to) a spectral polytope?No condition in Question 6.6 can be dropped. If we drop vertex-transitivity, wecould take some polytope whose edge-graph has trivial symmetry and only smalleigenspaces. Dropping ( ii ) leaves vertex-transitive neighborly polytopes, for whichwe know that these are mostly not spectral (except for the simplex). Dropping ( iii ) leaves us with the prisms and anti-prisms, the eigenspaces of their edge-graphs arerarely of dimension greater than two.Finally, we wonder whether these spectral techniques can be any help in classi-fying the edge-transitive polytopes. Question 6.7.
Can we classify the edge-transitive graphs that are spectral, and bythis, the edge-transitive polytopes?
Question 6.8.
Can the existence of half-transitive polytopes be excluded by usingspectral graph theory (see Section 5.3)?
Acknowledgements.
The author gratefully acknowledges the support by thefunding of the European Union and the Free State of Saxony (ESF).
IGENPOLYTOPES, SPECTRAL POLYTOPES AND EDGE-TRANSITIVITY 21
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Appendix A. Implementation in Mathematica
The following short Mathematica script takes as input a graph G (in the examplebelow, this is the edge-graph of the dodecahedron), and an index k of an eigenvalue.It then compute the v i (or vert in the code), i.e., the vertex-coordinates of the θ k -eigenpolytope. If the dimension turns out to be appropriate, the spectral embeddingof the graph, as well as the eigenpolytope are plotted. (* Input :* the graph G , and* the index k of an eigenvalue (k = 1 being the largest eigenvalue ).*)G = GraphData [" DodecahedralGraph " ];k = 2;(* Computation of vertex coordinates ’vert ’ *)n = VertexCount [G ];A = AdjacencyMatrix [G ];eval = Tally [ Sort @ Eigenvalues [A // N], Round [ θ ," mult "}} , eval ], Frame → All ]Which [d <2 , Print [" Dimension too low , no plot generated ."],d ==2 , GraphPlot [G , VertexCoordinates → vert ],d ==3 , GraphPlot3D [G , VertexCoordinates → vert ,d >3 , Print [" Dimension too high , 3- dimensional projection is plotted ." ];GraphPlot3D [G , VertexCoordinates → vert [[;; ,1;;3]] ]]If [d ==2 || d ==3 ,Region ‘ Mesh ‘ MergeCells [ ConvexHullMesh [ vert ]]] Faculty of Mathematics, University of Technology, 09107 Chemnitz, Germany
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