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Spectral Realizations of Symmetric Graphs, Spectral Polytopes and Edge-Transitivity

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Faculty of Mathematics Professorship for Algorithmic and Discrete Mathematics Spectral Realizations of Symmetric Graphs, Spectral Polytopes and Edge-Transitivity Dissertation submitted to the Faculty of Mathematics at Technische Universität Chemnitz in accordance with the requirements for the degree doctor rerum naturalium (Dr. rer. nat.) by MSc. Martin Winter Chemnitz, April 26, 2021 Referees: Prof. Dr. Christoph Helmberg (supervisor) Prof. Dr. Michael Joswig Prof. Dr. Egon Schulte Winter, Martin Spectral Realizations of Symmetric Graphs, Spectral Polytopes and Edge-Transitivity Dissertation, Faculty of Mathematics Technische Universität Chemnitz, 2021 Bibliographical description Martin Winter Spectral Realizations of Symmetric Graphs, Spectral Polytopes and Edge-Transitivity Dissertation, 184 pages, Technische Universität Chemnitz, Faculty of Mathematics, 2021 Abstract A spectral graph realization is an embedding of a finite simple graph into Euclidean space that is constructed from the eigenvalues and eigenvectors of the graph’s adjacency matrix. It has previously been observed that some polytopes can be reconstructed from their edge-graphs by taking the convex hull of a spectral realization of this edge-graph. These polytopes, which we shall call spectral polytopes, have remarkable rigidity and symmetry properties and are a source for many open questions. In this thesis we aim to further the understanding of this phenomenon by exploring the geometric and combinatorial properties of spectral polytopes on several levels. One of our central questions is whether already “weak” forms of symmetry can be a sufficient reason for a polytope to be spectral. To answer this, we derive a geometric criterion for the identifica- tion of spectral polytopes and apply it to prove that indeed all polytopes of combined vertex- and edge-transitivity are spectral, admit a unique reconstruction from the edge-graph and realize all the symmetries of this edge-graph. We explore the same questions for graph real- izations and find that realizations of combined vertex- and edge-transitivity are not necessar- ily spectral. Instead we show that we require a stronger form of symmetry, called distance- transitivity. Motivated by these findings we take a closer look at the class of edge-transitive polytopes, for which no classification is known. We state a conjecture for a potential classification and pro- vide complete classifications for several sub-classes, such as distance-transitive polytopes and edge-transitive polytopes that are not vertex-transitive. In particular, we show that the latter class contains only polytopes of dimension d 3. As a side result we obtain the complete classification≤ of the vertex-transitive zonotopes and a new characterization for root systems. Keywords Spectral graph realizations, eigenpolytopes, spectral polytopes, edge-transitive polytopes, vertex-transitive zonotopes 3 Acknowledgments Foremost, I am extremely grateful to my doctoral advisor Prof. Dr. Christoph Helmberg, who gave me the freedom to follow my interests, supported me in every instance, read all my papers, improved them by his detailed comments and was always available for questions. I also thank Dr. Thomas Jahn and Dr. Frank Göring for many hours of fruitful discussions. As a result of one such discussion Frank came up with an ingenious argument to significantly shorten the final step in one of the classification proofs presented in this thesis. Special thanks go to Prof. Dr. Achill Schürmann and Prof. Dr. Raman Sanyal who provided valuable feedback and put me on the right track with their literature references. Thanks go also to my numerous proofreaders (foremost Christin, Daniela, Marvin, Thomas) who made my work more readable with every hint after I became blind to the shortcomings myself. Finally, I would like to thank Prof. Dr. Michael Joswig and Prof. Dr. Egon Schulte for the time and effort they invested in reviewing this thesis. The author also gratefully acknowledges the support by the funding of the European Union and the Free State of Saxony who enabled this project in the first place. 5 Contents List of Symbols 11 Introduction 13 I Spectrum and Symmetry 25 1 Point Arrangements 27 1.1 The arrangement space . 28 1.2 Symmetric arrangements . 32 1.3 Separating symmetries . 37 1.4 Deformation and rigidity . 40 1.5 Transitive arrangements . 45 2 Graph Realizations 51 2.1 Spectral and balanced realizations . 53 2.2 Edge- and arc-transitive realizations . 58 2.3 Full local dimension . 62 2.4 Distance-transitive realizations . 65 2.5 Cosine vectors and cosine sequences . 67 3 Eigenpolytopes and Spectral Polytopes 73 3.1 Eigenpolytopes . 74 3.2 Spectral polytopes . 76 3.3 The Theorem of Izmestiev . 80 II Transitivities in Convex Polytopes 85 4 Edge-Transitive Polytopes 87 4.1 An overview of transitivity in polytopes . 88 4.2 Known edge-transitive polytopes . 89 4.3 Simultaneously vertex- and edge-transitive polytopes . 91 4.4 Arc-transitive polytopes . 93 4.5 Half-transitive polytopes . 97 4.6 Distance-transitive polytopes . 98 5 Vertex-Transitive Zonotopes 101 5.1 Generators, faces, symmetry and permutahedra . 102 7 Contents 5.2 The case d = 2....................................... 106 5.3 The general case . 108 5.4 The classification . 110 5.5 Related topics . 112 6 Purely Edge-Transitive Polytopes 115 6.1 Bipartite polytopes . 116 6.2 Some easy cases . 119 6.2.1 Bipartite polygons . 119 6.2.2 Inscribed bipartite polytopes . 120 6.2.3 Strictly bipartite polytopes for d 4..................... 121 6.3 Strictly bipartite polyhedra . .≥ . 121 6.4 Adjacent pairs . 128 6.4.1 The case τ1 = (4, 6, 10) ............................. 130 6.4.2 The case τ1 = (4, 6, 8) .............................. 131 6.4.3 The case τ1 = (4, 6, 6) .............................. 131 6.4.4 The case τ1 = (4, 4, 4) .............................. 132 6.4.5 The case τ1 = (4, 4, 2k), 2k 6........................ 133 ≥ 6.4.6 The case τ1 = (4, 4, 6) .............................. 135 6.5 An almost strictly bipartite polyhedron . 135 Conclusions and Outlook 141 Theses 145 Appendix 147 A Matrix Groups and Representations 149 A.1 Matrix groups . 149 A.2 Representations . 151 B Spectral Graph Theory 153 C Polytope Theory 155 C.1 Zonotopes . 156 D Reflection Groups and Root Systems 161 D.1 Reflection groups . 161 D.2 Root systems . 162 D.3 Coxeter-Dynkin diagrams . 162 E Wythoffian, Uniform and Regular Polytopes 165 E.1 Orbit polytopes and Wythoffian polytopes . 165 E.2 Coxeter-Dynkin diagrams for Wythoffian polytopes . 165 E.3 Uniform and regular polytopes . 166 8 Contents F Mathematica Scripts 169 G Additional proofs and computations 171 G.1 Two geometric statements . 171 G.2 A proof for spherical interior angles . 172 9 List of Symbols d R d-dimensional Euclidean space dim dimension of linear or affine space span linear span/hull of a set of vectors aff affine hull of a set of points conv convex hull of a set of points rank rank of a matrix or set of vectors, equivalent to dim span Id identity matrix (of appropriate dimension) d d d GL(R ) group of invertible linear transformations R R d d d! O(R ) group of orthogonal transformations R R d d ! d SO(R ) group of orthogonal transformations R R with determinant one ! πW orthogonal projection onto the subspace W deg(i) vertex degree of the vertex i deg(G) (common) vertex degree of the vertices in a regular graph G dist(i, j) graph theoretic distance between two vertices i, j V diam(G) diameter of the graph G 2 Aut( ) automorphism group of a point arrangement, graph, polytope, etc. · GP edge-graph of the polytope P d skP skeleton of P, a graph realization skP : V (GP ) R of the edge-graph F(P) face lattice of the polytope P ! Fδ(P) set of δ-dimensional faces of the polytope P Zon(R) zonotope generated by R Gen(Z) (standard) generators of the zonotope Z Sym(V ) symmetric group on V , i.e., the group of permutations of the set V n n Perm(R ) group of permutation matrices on R Πσ permutation matrix associated to the permutation σ id identity permutation d Ad (Σ) space of full-dimensional Σ-arrangements in R d Ad (G, Σ) space of full-dimensional Σ-realizations of G in R Spec(G) (adjacency) spectrum of of the graph G EigG(θ) θ-eigenspace of the (adjacency matrix of the) graph G PG(θ) θ-eigenpolytope of the graph G θ eigG θ-realization used to construct the θ-eigenpolytope x Euclidean norm of a vector x k k d x, y standard inner product in R h i Ý(x, y) angle between the vectors x and y d d 1 S d-dimensional sphere (embedded in R + ) 11 List of Symbols Sr (c) sphere with center c and radius r (of appropriate dimension) d vol (relative) volume of a subset of R 12 Introduction The product of mathematics is clarity and understanding. Not theorems, by themselves. William Thurston The content of this thesis was developed in an effort to understand a series of curious ob- servations made while working on so-called spectral graph realizations. Those are a technique to embed a graph (a purely combinatorial object) into Euclidean space by using the eigenval- ues and eigenvectors of its adjacency matrix. Applying this to the edge-graphs of polytopes results in some cases in an embedding that looks just like the skeleton of the original poly- tope. This phenomenon points to connections between polytope theory and spectral graph theory, between convexity and rigidity, between symmetry in geometry and combinatorics, all to be explored in this thesis. A curious observation The best way to motivate the investigation of the described phenomenon is to see it happen. Consider the 3-dimensional cube and let G = (V, E) be its edge-graph with vertex set V = 1, ..., 8 , numbers assigned to the vertices according to the following figure: f g 8 8 To G we can assign a matrix A 0, 1 × , the so-called adjacency matrix, with Ai j = 1 if and only if i j E is an edge.
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    J. Korean Math. Soc. 49 (2012), No. 6, pp. 1111–1121 http://dx.doi.org/10.4134/JKMS.2012.49.6.1111 TETRAVALENT SYMMETRIC GRAPHS OF ORDER 9p Song-Tao Guo and Yan-Quan Feng Abstract. A graph is symmetric if its automorphism group acts transi- tively on the set of arcs of the graph. In this paper, we classify tetravalent symmetric graphs of order 9p for each prime p. 1. Introduction Let G be a permutation group on a set Ω and α ∈ Ω. Denote by Gα the stabilizer of α in G, that is, the subgroup of G fixing the point α. We say that G is semiregular on Ω if Gα = 1 for every α ∈ Ω and regular if G is transitive and semiregular. Throughout this paper, we consider undirected finite connected graphs without loops or multiple edges. For a graph X we use V (X), E(X) and Aut(X) to denote its vertex set, edge set, and automorphism group, respectively. For u , v ∈ V (X), denote by {u , v} the edge incident to u and v in X. A graph X is said to be vertex-transitive if Aut(X) acts transitively on V (X). An s-arc in a graph is an ordered (s + 1)-tuple (v0, v1,...,vs−1, vs) of vertices of the graph X such that vi−1 is adjacent to vi for 1 ≤ i ≤ s, and vi−1 = vi+1 for 1 ≤ i ≤ s − 1. In particular, a 1-arc is called an arc for short and a 0-arc is a vertex.
  • Constructing Arbitrarily Large Graphs with a Specified Number Of

    Constructing Arbitrarily Large Graphs with a Specified Number Of

    Electronic Journal of Graph Theory and Applications 4 (1) (2019), 1–8 Constructing Arbitrarily Large Graphs with a Specified Number of Hamiltonian Cycles Michael Haythorpea aSchool of Computer Science, Engineering and Mathematics, Flinders University, 1284 South Road, Clovelly Park, SA 5042, Australia michael.haythorpe@flinders.edu.au Abstract A constructive method is provided that outputs a directed graph which is named a broken crown graph, containing 5n − 9 vertices and k Hamiltonian cycles for any choice of integers n ≥ k ≥ 4. The construction is not designed to be minimal in any sense, but rather to ensure that the graphs produced remain non-trivial instances of the Hamiltonian cycle problem even when k is chosen to be much smaller than n. Keywords: Hamiltonian cycles, Graph Construction, Broken Crown Mathematics Subject Classification : 05C45 1. Introduction The Hamiltonian cycle problem (HCP) is a famous NP-complete problem in which one must determine whether a given graph contains a simple cycle traversing all vertices of the graph, or not. Such a simple cycle is called a Hamiltonian cycle (HC), and a graph containing at least one arXiv:1902.10351v1 [math.CO] 27 Feb 2019 Hamiltonian cycle is said to be a Hamiltonian graph. Typically, randomly generated graphs (such as Erdos-R˝ enyi´ graphs), if connected, are Hamil- tonian and contain many Hamiltonian cycles. Although HCP is an NP-complete problem, for these graphs it is often fairly easy for a sophisticated heuristic (e.g. see Concorde [1], Keld Helsgaun’s LKH [4] or Snakes-and-ladders Heuristic [2]) to discover one of the multitude of Hamiltonian cy- cles through a clever search.