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CONFIGURATIONSFrom a graphical viewpoint
Tomaz Pisanski1
IMFMOddelek za teoreticno racunalnistvo
Univerza v LjubljaniJadranska 19, 1111 Ljubljana
andBrigitte Servatius
WPIUSA
Copyright c©, Tomaz Pisanski, Brigitte Servatius 1999–2004Ljubljana, 2004
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The first author used the notes as a reading material for students of agraduate course Konfiguracije in grafi (Configurations and Graphs) at theUniversity of Ljubljana, Slovenia, 1999–2000.
Several people contributed to these notes. Aleksandar Jurisic suggestedseveral important references. Several co-authors allowed him to include partsof our joint work verbatim in the earlier versions of the manuscript. Weare grateful to Anton Betten, Marko Boben, Gunnar Brinkmann, MarstonConder, Milan Hladnik, Ante Graovac, Harald Gropp, Aleksander Malnic,Dragan Marusic, Alen Orbanic, Marko Petkovsek, Primoz Potocnik, MilanRandic, Steve Wilson, and Arjana Zitnik, for that. We would like to thankMarko Boben, Jurgen Bokowski, Chris Godsil, Josef Leydold, Alen Orbanic,Marko Petkovsek, Bor Plestenjak, John Shawe-Taylor, for reading severalversions of the manuscript or prats of the manuscript, giving some veryinteresting suggestions.
After finishing teaching the course T.P. ceased working on the manuscriptfor about seven months. When he started preparing these notes he was un-aware of some of the most recent work on configurations of Branko Grunbaum.We would like to thank him for sharing his work with us. After receiving hisnotes for a graduate course on configurations [45] T.P. resumed his work onthe manuscript in January 2001 and began adjusting terminology with thatof Grunbaum in all feasible cases.
In 2001-2003 he used part of these notes for another graduate course onconfigurartions for mathematics teachers.
In 2002-2003 Brigitte Servatius spent her sabbatical in Ljubljana. Wedecided to turn these notes into a book.
The Fall semester of 2003 T.P. spent at Colgate University which enabledfurther communication.
The work was greatly simplified by using crossplatform cvs system asproposed by Andrej Bauer. Herman Servatius helped us a lot not only bydrawing a number of figures but also with his expertise as a mathematicianand as expert latex user.
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Figure 1: Configurations lie in the heart of graphs, groups, sutfaces andgeometries.
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Foreword
When I first learned about sets in high school my view of a static mathematicscollapsed. In elementary school I was convinced that most mathematicswas known to the old Greeks and was sure that anything important wasdiscovered long time ago.
I was thrilled when I was told that sets are at the basis of modern mathe-matics and was shocked to learn that the set of all sets does not exist. Laterin college we were informed that categories are about to replace sets in thefoundations of mathematics. When I discovered for myself that graphs aremore general than categories I was glad that I have chosen such an importantconcept which is related to the roots of mathematics as the prime object ofmy studies.
Roughly and very informally speaking one can divide graphs just like realnumbers or most other mathematical concepts into two classes: there is alarge volume of dull graphs and a small portion of interesting graphs. Let ustry to clarify the classes of dull and intersting objects first with real numbers.How to distinguish interesting reals from the dull ones? Maybe we shoulddefine interesting reals as those real numbers that admit finite description.Since there are at most countably many finite descriptions, it is not hard tosee that the vast majority of reals are dull. For graphs we have to modifythis definition, otherwise all finite graphs would qualify as interesting. Anygraph on n vertices can be described by its adjacency matrix using about n2
bits. One can say that a graph is interesting if it admits a short description;for intstance, if it can be described by at most f(n) bits of information wheref is an increasing function but the ratio f(n)/n2 tends to 0 when n tends toinfinity.
The research in graph theory can be classified into two classes:
• Class I. Find interesting properties of (dull) graphs.
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• Class II. Study families of interesting graphs.
A typical result of Class I is the fact that there is no disconnected graphwhose complement is also disconnected. On the other hand a Class II resultis the fact that there are exactly seven arc-transitive generalized Petersengraphs. These notes would qualify under Class II philosophy. We not onlylook at some finite families of graphs but consider individual graphs withunique properties.
The final question. Why would one like to revive the interest in con-figurations, a subject that used to be very much alive in the second halfof the nineteenth century? A partial answer lies in the following analogy.Graphs are used to model other mathematical structures. Such graphs re-flect properties of the structures they model and this makes them sometimesvery interesting. For instance groups are modeled by their Cayley graphs.In a similar way geometries and configurations can be modeled by their Levigraphs. That is why we may expect that several configurations will give riseto some very interesting graphs too.
The more I wrote these notes the more it became obvious to me that wecan paraphrase the key moto of Art White’s book [82] by putting configura-tions in the centre of a tetrahedron whose four vertices are:
1. Graphs - Basic Graph Theory
2. Groups - Algebraic Graph Theory
3. Surfaces - Topological Graph Theory
4. Geometries - Geometrical Graph Theory
see Figure 1.This text can be regarded as an ad hoc introduction to the interesting
interplay of these topics. The reader will soon find out that configurationsplay a catalytic role in this book and though considered as the principal actorthey do not appear in every scene.
Tomaz Pisanski, Ljubljana, 2002.
Table of Contents
Foreword 5
Table of Contents 70.1 Permutations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1 FROM GRAPHS TO GEOMETRIES 191.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.1.1 Examples of Graphs . . . . . . . . . . . . . . . . . . . 201.1.2 Isomorphism and Invariants. . . . . . . . . . . . . . . . 331.1.3 Skeleta. . . . . . . . . . . . . . . . . . . . . . . . . . . 331.1.4 Operations on graphs. . . . . . . . . . . . . . . . . . . 40
1.2 From Geometry to Graphs and Back . . . . . . . . . . . . . . 461.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2 FROM GROUPS TO SURFACES 632.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.2 Cayley Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 722.3 Symmetry and Orbits in Graphs . . . . . . . . . . . . . . . . . 752.4 Voltage Graphs and Covering Graphs. . . . . . . . . . . . . . 832.5 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.5.1 Operations on Maps . . . . . . . . . . . . . . . . . . . 1172.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3 INCIDENCE GEOMETRIES 1433.1 Incidence structures . . . . . . . . . . . . . . . . . . . . . . . . 1433.2 Incidence geometry . . . . . . . . . . . . . . . . . . . . . . . . 1453.3 Incidence Structures Revisited . . . . . . . . . . . . . . . . . . 160
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8 TABLE OF CONTENTS
3.4 Projective and Affine Planes and Spaces . . . . . . . . . . . . 1703.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4 CONFIGURATIONS 1834.1 Classical configurations . . . . . . . . . . . . . . . . . . . . . . 1884.2 Self-polar Combinatorial Configurations . . . . . . . . . . . . . 2004.3 Generalized Petersen Graphs and Haar Graphs . . . . . . . . . 2034.4 The remarkable G(8, 3) . . . . . . . . . . . . . . . . . . . . . . 2074.5 Bicayley Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 2114.6 Cyclic configurations . . . . . . . . . . . . . . . . . . . . . . . 216
4.6.1 Cyclic configurations and Haar graphs . . . . . . . . . 2164.7 Polycyclic Configrations - Update with new version needed . . 2184.8 Weakly flag-transitive configurations . . . . . . . . . . . . . . 2334.9 Representation of combinatorial configurations . . . . . . . . . 242
4.9.1 Coordinatization . . . . . . . . . . . . . . . . . . . . . 2444.9.2 The Steinitz Theorem . . . . . . . . . . . . . . . . . . 2464.9.3 Some non-realizable (v3) configurations . . . . . . . . . 2464.9.4 The flag-sum of two (v3) configurations . . . . . . . . . 2544.9.5 In general ρ 6= δ . . . . . . . . . . . . . . . . . . . . . . 255
4.10 Exercises Chapter IV . . . . . . . . . . . . . . . . . . . . . . . 257
General Index 263
References 272
INTRODUCTION
§ 1. Hexagrammum mysticum. In 1640, two years before inventingthe first mechanical calculator, Blaise Pascal (1623 – 1662) was 16 years oldwhen he published a small pamphlet entitled Essai pour les coniques whichcontains the following remarkable theorem.
Figure 2: Hexagrammum mysticum.
Theorem 0.1 (Blaise Pascal, 1639). If a hexagon is inscribed in a conic,then the three points in which pairs of opposite sides meet will lie on a straightline.
It seems that Pascal proved the theorem in June 1639. According toCoxeter [18] G. W. Leibniz (1646 – 1716) admired Pascal’s proof when visitingParis. Unfortunately the proof was later lost.
The line whose existence is asserted by the above theorem is sometimescalled Pascal line. Six distinct objects can be arranged in circular orderin 60 different ways, so for a given set of 6 points on a conic there are 60hexagons, which in turn give rise to 60 Pascal lines. Several geometers were
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10 TABLE OF CONTENTS
attracted by these 60 lines. Kirkman, Cayley, Steiner, Plucker, and Salmonhave studied this figure or configuration which is called Pascal configurationby David Wells on page 172 of [81]. H. F. Baker, Sc.D., LL.D., F.R.S.,Lowdean Professor of Astronomy and Geometry, and Fellow of St. John’sCollege, at the University of Cambridge provides a list of references to thismatter in the second volume of his 6-volume opus [5]. The list includesthe following contributors: Pascal(1640), Brianchon (1806), Steiner (1828),Plucker (1830), Hesse (1842), Cayley (1846), Grossman (1861), Von Staudt(1863), Bauer (1874), Kirkman (1849), Veronese (1877), Cremona (1877),Sylvester(18??), Salmon (1879), Klug (1898), Castelnuovo (1887), Caporali(1888), and Richmond (1894). Baker calls the Pascal configuration also Hex-agrammum mysticum. In addition to 60 Pascal lines it contains 20 Steinerpoints, 60 Kirkman points, 20 Cayley lines, 15 Plucker lines and 15 Salmonpoints. It is possible to draw a picture of Hexagrammum mysticum, howeverit is very hard to see the relationships that hold among its various types ofpoints and lines.
One of the purposes of these notes is to enable the reader to see the Pascalconfiguration from a different view-point using graphs. The emphasis is thusshifted from geometry to combinatorics. In the first book on configurations,[51], Friederich Levi uses about 50 pages of Chapter 5 in order to study theproperties of Pascal’s configuration.
§ 2. Why configurations? The topic of configurations seems to beworth studying for various reasons. Many great mathematicians contributedto the development of the theory. Configurations lie at the intersection of var-ious mathematical disciplines. Nowadays they form a part of combinatoricsbut their origin lies in geometry of the 19th century. In particular, curvesand surfaces of the projective geometry give rise to some important histor-ical configurations. Configurations can be studied from a graph-theoreticalviewpoint via the so-called Levi graphs. At the same time they bring geo-metrical flavor to graphs. Symmetries play an important role in the studyof configurations. Recently several new problems have arisen that put con-figurations in the domain of theoretical computer science. The theory hasevolved from various sources and viewpoints which makes the objects of ourinterest visible in quite diverse lights.
The central theme of modern combinatorics is probably the concept ofcombinatorial design which is in our view too narrow a structure. If we
TABLE OF CONTENTS 11
want to embrace all configurations that were studied in the past we have toconsider more general combinatorial structures. Perhaps the most generalstructure that is available is the idea of geometry in the sense of Tits [74] .
It covers all structures that are needed in these notes. Although it rep-resents a deep and powerful tool in connection with groups it is by itselfnothing but a graph endowed with a vertex coloring and is readily availableto anyone with basic knowledge of combinatorics and graph theory.
The fundamental question about configurations, that is, the questionabout their existence was posed by T. Reye [66]. This question has severalrefinements. For instance one can ask not only which combinatorial con-figurations exist but also which geometric configurations of points and linesexist. A more complete answer is provided with the list of all non-isomorphicconfigurations of a given type. Quite recently Branko Grunbaum [46] gave asurprising answer to the problem of configurations for the symmetric case, byproving that for any set of large parameters one can construct a symmetricgeometric configuration with those parameters.
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 98 7 2 1 6 9 3 4 56 1 8 9 7 3 4 5 2
Figure 3: Configuration table for the Pappus configuration.
§ 3. Examples. Let us give several examples of configurations in orderto motivate the reader.
12 TABLE OF CONTENTS
Example 0.2. Figure 3 depicts a configuration of points and lines, knownas the Pappus configuration.
1
2
3
4
5
6
7
1 2 3 4 5 6 75 7 2 1 3 4 66 5 1 7 4 2 3
Figure 4: Combinatorial configuration (73), the Fano plane.
Example 0.3. Figure 4 depicts a configuration known as the Fano configu-ration. It is defined by its configuration table and can be represented in theordinary Euclidean plane by 7 points, 6 lines and one curve.
Example 0.4. Figure 5 depicts a configuration known as the Miquel config-uration. It consists of 8 points, 6 circles (and lines).
TABLE OF CONTENTS 13
1 3 1 5 1 22 4 2 6 3 45 7 3 7 5 66 8 4 8 7 8
Figure 5: Miquel configuration of points andcircles.
Although Miquel configuration is ”geometric” in the sense that is definedby geometric objects and their incidence, it differs from the previous twoexamples: it cannot be defined as a configuration of points and lines sincetwo circles may intersect more than once! Its designation is (83, 64)2.
Example 0.5. Finally, consider the tetrahedron depicted in Figure 6 withvertices denoted by A,B,C,D. We may denote the edges by 1 = AB, 2 =AC, 3 = AD, 4 = BC, 5 = BD, 6 = CD and faces by α = BCD, β =ACD, γ = ABD, δ = ABC.
A
B
C
D
Figure 6: The tetrahedron gives rise to six configurations.
14 TABLE OF CONTENTS
We can consider configurations defined by
• vertices and edges,
• vertices and faces,
• edges and faces,
• edges and vertices,
• faces and vertices,
• faces and edges.
The situation can be described in a tabular form.
v = 4 e = 6 f = 4v = 4 0 3 3e = 6 2 0 2f = 4 3 3 0
For instance, entry 2 in the second row (e = 6) and first column (v = 2)tells us that there are 2 vertices at each edge.
Here is the configuration table for the configuration of points and faces
α β γ δ
B A A AC C B BD D D C
0.1 Permutations.
We would like to recall a few facts about permutations. A permutation onthe set V is a bijection π : V → V of V on itself. The set of all permutationson V is denoted by Sym(V ). Usually we consider the “standard set” V ={1, 2, ..., n}. In this case we write Sym(V ) = Sym(n) = Sn. Usually we writea permutation in the form
π =
(v1 v2 · · · vn
π(v1) π(v2) · · · π(vn)
).
0.1. PERMUTATIONS. 15
If we have the standard set, the notation can be shortened and only thesecond row is given.
π =[π(1) π(2) · · · π(n)
].
Example 0.6. Each row of the configuration table of Pappus configurationcan be viewed as a permutation.
α1 =
(1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9
).
α2 =
(1 2 3 4 5 6 7 8 98 7 2 1 6 9 3 4 5
).
α3 =
(1 2 3 4 5 6 7 8 96 1 8 9 7 3 4 5 2
).
Recall that each permutation can be written in a unique way as a productof disjoint cycles:
α1 = (1)(2)(3)(4)(5)(6)(7)(8)(9)
α2 = (184)(273)(569)
α3 = (16385749)
Permutation consisting of a single cycle is called cyclic permutation.Hence α3 is cyclic permutation. If all cycles are of the same length, thepermutation is called polycyclic. If all cycles are of length 1, the permutationis called the identity permutation. An element x ∈ V for which π(x) = x iscalled a fixed point of π. Let Fix(π) denote: Fix(π) = {x ∈ V |π(x) = x}and let fix(π) = |Fix(π)|. Hence fix(α1) = 9, f ix(α2) = fix(α3) = 0. Afixed-point free permutation is also called a derangement. The set of derange-ments over V is denoted by Der(V ). If seen as a subset of Sn it is denoted byDn. A permutation whose longest cycle has length 2 is called an involution.If, in addition, it has no fixed points, it is called a fixed-point free involution.
Each permutation can be depicted in graphic form. Each element x ∈ Vis represented by a little circle and we draw an arrow from x to π(x).
16 TABLE OF CONTENTS
α1 α2 α3
Figure 7: Permutations α1, α2, α3.
0.2. EXERCISES 17
0.2 Exercises
For a warm-up we give the following problems. Problems denoted by (*) aremore difficult and usually require material that was not covered up to thispoint. Problems denoted by (**) represent research problems or very difficultproblems.
Exercise 0.1. How many points and lines does the Pappus configurationhave?
Exercise 0.2. How many lines pass through a given point of the Pappusconfiguration?
Exercise 0.3. How many points lie on a given line of the Pappus configura-tion?
Exercise 0.4. What is the minimum and what is the maximum number oflines passing through a pair of distinct points of the Pappus configuration?
Exercise 0.5 (*). Plant 9 trees in 10 rows in such a way that there are 3trees in each of the 10 rows.
Exercise 0.6 (*). Prove Pascal’s Theorem for the case when the conic iscircle.
Exercise 0.7 (*). Derive the Pappus configuration from Pascal’s Theoremby considering two lines as a degenerate conic.
Exercise 0.8. Determine the lines that pass through the point labelled 3 ofthe Fano configuration.
Exercise 0.9. How many lines pass through points 4 and 6 of the Fanoconfiguration?
Exercise 0.10. How many lines pass through points 1,2, and 3 of the Fanoconfiguration?
Exercise 0.11 (*). Is it possible to draw Fano’s configuration with straightlines?
Exercise 0.12. Write down the configuration tables for the six configura-tions determined by the tetrahedron. Verify that the point-line and face-lineconfigurations have essentially the same configuration table. Verify that thepoint-face and face-point configuration tables are essentially the same.
18 TABLE OF CONTENTS
Exercise 0.13. In a presidential election in some small country there are 8presidential candidates. Schedule a series of 8 TV debates in such a way, thatthere are three presidential candidates in each debate and no two candidatesmeet in the studio more than once. Show that the solution is essentiallyunique.
Exercise 0.14. Write down all permutations from S3.
Exercise 0.15. Write down all involutions from S4.
Exercise 0.16. Write down all fixed-point free permutations from S5. Inother words, determine the set D5.
Exercise 0.17. Write down all fixed-point free involutions from S4 and fromS5.
Exercise 0.18. A permutation is called semi-regular, if it is a product ofcycles of the same size. Determine the number of semi-regular permutationsfrom S2005.
Exercise 0.19. Let p be a prime. Show that the number of semi-regularpermutations from Sp equals (p− 1)!
Exercise 0.20. (*) Determine the number of derangements in Dn.
Exercise 0.21. (*) Determine the number of involutions from Sn.
Exercise 0.22. (*) Determine the number of fixed-point free involutions fromSn.
General Index
(d, g)-cage, 931/2-arc-transitive, 79103 anti-pappian configuration, 251t-arc, 78t-arc-transitive, 78Adam conjecture, 213Zitnik, Arjana, 2Skoviera, Martin, 821-dimensional subdivision, 1251-factor, 442-dimensional subdivision, 1292-skeleton, 973-arc-transitive bipartite graph, 2576-cage, 1888-cage, 91
absolute line, 200absolute point, 200abstract polytope, 154abstract simplicial complex, 152addition, 143adjacency, 19affine plane, 172affine space, 172alternating group, 70Antipodal dual, 131archimedean graphs, 36Archimedean polyhedra, 75augmented plane, 171automorphism, 63
Baker,, 159Baker, Henry, 10Balaban,Alexandru, 31ball closed, 47barycentric subdivision, 129base graph, 90Bauer, , 10Bauer, Andrej, 2benzenoid graph, 58Betten, Anton, 2bi-Cayley graph, 211bicirculant, 211Biggs, Norman, 78binary tree, 29bipartition set, 82block system, 184Boben, Marko, 2, 176Bokowski, Jurgen, 2bouquet of circles, 123Bouwer, , 81Brianchon, Charles, 10Brinkmann, Gunnar, 2Burnside’s Lemma, 74
cage, 30canonical nuber, 214Caporali, , 10cartesian product, 40Castelnuovo, Guido, 10Cayley graph, 72, 138, 211Cayley line, 10
263
264 GENERAL INDEX
Cayley, Arthur, 10center, 100chamber, 152, 156chamber graph, 152, 156, 157chamber system, 156chamfering operation, 133chromatic number, 53circulant, 211
symbol, 211clique, 151closed ball, 47co-flag, 163combinatorial configuration, 11combinatorial design, 10combinatorial map, 129, 157combinatorial surface, 108commutator, 72commute, 65complete bipartite graph, 25complete graph, 23complete multipartite graph, 26complex number, 68Conder, Marston, 2cone, 43
apex, 43configuration, 10, 12, 183configuration table, 216configurations
representation, 242conjugate, 66connected, 29connectivity, 29convex closure, 49convex polyhedron, 95convex set, 49covering graph, 90covering transformation, 139Coxeter graph, 196, 258
Coxeter, H. S. M. , 9Cremona, Luigi, 10, 196Cremona-Richmond configuration,
196Cremona-Richmond graph, 30crosscap, 108crosscap number, 97cube, 125cubic, 38cuboctahedron, 97cubohemioctahedron, 97cvs, 2cycle, 20cyclic, 216cyclic equivalence, 213cyclic Haar graph, 211cyclic permutation, 15
Durer graph, 29, 191Daublebsky von Sterneck, 194derangement, 15Desargues configuration, 192Desargures configuration, 159Descartes, Rene, 143design, 167diagram, 157dihedral group, 71dipole, 91, 211disk, 97distance, 47distance function, 47distance matrix, 47distance sequence, 82dodecahedron, 36double torus, 112dual, 123dual map, 104dual type, 146
GENERAL INDEX 265
embedded square, 132endpoint, 19endvertex, 19, 20Escher,M.C., 75Euclid, 143Euclidean geometry, 143Euler characteristic, 32Euler characteristics, 107Eulerian surface triangulation, 148
face, 32factorization, 156Fano configuration, 12Fano plane, 188Fano, Gino, 12fixed point, 15fixed-point free involution, 15fixed-point free permutation, 15flag, 101, 151
co-rank, 146, 156elliptic, 163hyperbolic, 163parabolic, 163rank, 146type, 146
flag system, 149Folkman graph, 81forest, 29four dimensional cube Q4, 191Frucht, Roberto, 82fullerene, 59
gallery, 156generalized k-gon, 157generalized n-gon, 167generalized digon, 180generalized Petersen graph, 87, 90generalized quadrangle, 157, 181
generalized Reye geometry, 155generalized rotagraph, 93generalized triangle, 180generator, 71genus, 96, 109geometric configuration, 11, 242geometric strongly regular graph,
167geometry
truncation, 152girth, 29Godsil, Chris, 2, 101Goldberg operations, 133Grunbaum graph, 147Grunbaum, Branko, 2, 11Graovac, Ante, 2graph, 19
K2,2,2, 953-connected, 95antiprism, 36, 93arc-transitive, 78automorphism, 63, 75clique, 146complement, 162cone, 43edge set, 19edge-orbit, 77edge-transitive, 77generalized Petersen graph, 27indistinguishable vertices, 75isomorphism, 33LCF notation, 38Moebius ladder, 55multigraph, 32polyhedral, 95pregraph, 33prism, 36, 93regular, 167
266 GENERAL INDEX
semi-edge, 32semi-symmetric, 81simple, 19strongly regular, 167vertex set, 19vertex-transitive, 77wheel, 38zero-symmetric, 80
graph complement, 42graph induced by polarity, 202graph join, 43graph representation, 51graph square, 43graph union, 42Graphical regular representation, 73graphical simplicial complex, 156Graver,, 82Gray graph, 81Gray, Marion C., 81Gropp, Harald, 2, 196Grossman, , 10group, 63, 64
Abelian, 66action, 69commutative, 66inverse, 65order, 64unit element, 65
group Θ, 103group A, 103group action
regular, 79group homomorphism, 66group isomorphism, 67group of covering transformations,
139group presentation, 71group table, 67
GRR, 73GRR (graphical regular represen-
tation, 211
Haar equivalence, 213Haar graph, 211Haar integral, 211halfedge, 90Hamilton cycle, 82hamilton cycle, 29Hamilton, , 82handcuff graph, 87handle, 108Handshaking Lemma, 19Hausdorff space, 100Heawood graph, 30, 188heptahedron (see tetrahemihexahe-
dron), 96Hess configuration, 157Hess, Edmund (1843–1903), 155, 157Hesse, Otto, 10hexagrammum mysticum, 9, 10, 30
Cayley-Salmon, 160Cremona point, 160Kirkman plane, 160Pascal line, 160Plucker point, 160Salmon plane, 160Steiner plane, 160
hexahedron, 36Hilbert,David, 143Hladnik, Milan, 2Hoelder,, 64homeomorphism, 100hypercube, 40, 152hyperplane, 152
icosahedron, 36
GENERAL INDEX 267
identity permutation, 15imprimitive group action, 69incidence geometry, 145
antiautomorphism, 146automorphism, 146
flag-transitive, 146maximal-flag-transitive, 146
chamber-transitive, 156configuration, 146connected, 146duality, 146flag, 146Levi graph, 146maximal flag, 146morphism, 146polarity, 146rank, 146residually connected, 151strongly connected, 147type, 146type function, 146
incidence geometry., 54incidence relation, 144incidence structure, 144
block graph, 162blocking set, 162blocking set free, 162co-flag, 161co-simple, 162colinear points, 162collinear points, 161configuration graph, 162connected, 162decomposable, 162degree of a block, 162degree of a block set, 162degree of a point, 162degree of a point set, 162
dual, 161dual Menger graph, 162dual structure, 161flag, 161incidence graph, 162incidence matrix, 161Levi graph, 162Menger graph, 162parallel class, 162partial parallel class, 163point graph, 162representation, 242resolvable, 163simple, 162square, 162symmetric, 162
index, 66induced subgraph, 29integers, 64integers mod n, 65internal vertex, 20interval graph, 49inverse, 65involution, 15, 33
fixed-point free, 15isomorphic graphs, 19isomorphism, 33, 63
Jurisic, Aleksandar, 2
kernel, 67Kirkman point, 10Kirkman, Thomas, 10Klein 4-group, 67Klein bottle, 97, 112Klein, Felix, 196Klug, , 10Krausz partition, 157
268 GENERAL INDEX
Kronecker double cover, 91
LCF notation, 38, 138leapfrog, 132left coset, 66Leibniz, Gottfried W. , 9Levi graph, 144, 146, 257Levi,, 101Levi, Friedrich, 10Leydold, Josef, 2line, 12line graph, 42, 80, 138, 139, 157line segment, 242line-graph, 127lineal incidence structure, 165linear algebra, 143linear space, 165
nondegenerate, 166nontrivial, 166proper, 166restricted, 166
Malnic, Aleksander, 2manifold, 100map, 97, 100, 157
non-orientable, 103orientable, 103
map projection, 105Marusic, Dragan, 2McMullen, Peter, 154medial, 127Menger graph of a given type, 147Merniger, Markus, 93metric space, 46Miquel configuration, 12Miquel,, 12Moebius configuration, 191Moebius-Kantor configuration, 190
monograph, 93multigraph, 32
n-connected, 29near-pencil, 166, 170nearlinear space, 165Nedela, Roman, 82non-orientable genus, 96nonorientable genus, 109normal subgroup, 66
objectdull, 5interesting, 5
octahedron, 36, 95octahemioctahedron, 97one-skeleton, 95open disk, 100Orbanicc, Alen, 2orbit, 69, 75ordered geometry, 153ordinary voltage assignment, 90orientable genus, 109orientation class, 103
Pappus, 12Pappus configuration, 12, 15, 192,
257Pappus graph, 141, 192, 257parallel classes, 172parallel simplification, 125parallelism, 172parallelization, 125partial geometry, 166Pascal configuration, 10Pascal line, 9, 10Pascal, Blaise, 9, 10Pasini geometry, 147
residuum, 147
GENERAL INDEX 269
Pasini,, 147path, 20
endvertex, 20internal vertex, 20
permutation, 14even, 70fixed-point free, 15odd, 70semi-regular, 18
permutation derived graph, 90permutation voltage assignment, 90Petersen graph, 30Petkovsek, Marko, 2Petrie dual, 131Plucker line, 10Plucker, Julius, 10planar incidence structure, 165platonic graphs, 36Platonic polyhedra, 75Plestenjak, Bor, 2point, 12polar decomposition, 202polar structure, 202polycyclic permutation, 15polygonal complex, 149polyhedron, 95poset, 153
interval, 154Potocnik, Primoz, 2power set, 155pregraph, 33projective plane, 112, 157
topological, 96projective space, 171pseudo-geometric strongly regular
graph, 167pseudoline, 242pure graph square, 147
pyramid graph, 38
quaternion, 68quaternion group, 68quotient, 66
Randic, Milan, 2, 138rank function, 153ranked poset, 153
connected, 154path, 154
real numbers, 64real plane, 143reduced word, 71redundant number, 215regular map., 116regular polar decomposition, 202relator, 71representation, 242
flag-faithful, 243line-faithful, 243point-faithful, 243
residue, 151residuum, 151Reye configuration, 145Reye,Theodor, 11Richmond,, 196Richmond, Herbert, 10right coset, 66Ringel, Gerhard, 101rotagraph, 93rotation square, 132Royle, Gordon, 101
Sabidussi,Gert, 79Salmon pint, 10Salmon, Georg, 10Salmon, George, 10scalar multiplication, 143
270 GENERAL INDEX
Schlegel diagram, 95, 100Schur norm, 211Segre, B. (1942), 159semi-edge, 32, 211semi-regular, 45semicyclic configuration, 216semiedge (see halfedge), 90semilinear incidence structure, 165Servatius, Herman, 2Servatius,Brigitte, 2shadow, 152shadow space, 152Shawe-Taylor, John, 2Shulte, Egon, 154simple graph, 19simplicial complex, 155simplification, 125skeleton, 33, 46snub, 132spanning subgraph, 29spanning tree, 29, 32sphere, 97, 112Steiner point, 10Steiner, Jakob, 10Steinitz,Ernst, 159stereographic projection, 95strong product, 42strongly connected geometry, 147strongly flag-transitive configuration.,
233strongly regular graph, 167strongly regular polar decomposi-
tion, 202subdivision graph, 42subgeometry, 152subgraph
spanning, 29subgroup, 65
surface, 100suspension, 43Sylow subgroup, 138Sylow, , 138Sylvester, James, 10
tensor product, 40, 93ternary ring, 173tetrahedron, 13, 36, 123tetrahemihexahedron, 96tetravelent, 80theta graph, 91thick geometry, 156thin geometry, 156Tits,Jacques, 11Titts building, 151Titts, Jacques, 151torus, 97, 112transversal geometry, 156transversal line, 158tree, 29, 32triangle, 157truncated geometry, 152truncation, 129Tutte 8-cage, 30Tutte,, 101
uniform polyhedron, 96unit, 48unit distance graph., 51unit element, 65unit sphere graph, 48, 49unitary map, 108
valence, 19vector space, 143
addition, 143scalar multiplication, 143
Vega, 20
GENERAL INDEX 271
Veronese, Giuseppe, 10vertex coloring, 53, 155vertex figure, 153vertex orbit, 77voltage graph, 90Von Staudt, Karl, 10
wall, 152, 156Watkins graph, 261Watkins, Mark, 82, 261weakly flag-transitive configuration,
233Wells, David, 10White, Arthur, 6Wilson, Steve, 2word, 71
272 GENERAL INDEX
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