## Bridges between Geometry and Graph Theory (0)

Venue: | in Geometry at Work, C.A. Gorini, ed., MAA Notes 53 |

Citations: | 9 - 4 self |

### BibTeX

@INPROCEEDINGS{Pisanski_bridgesbetween,

author = {Tomaz Pisanski},

title = {Bridges between Geometry and Graph Theory},

booktitle = {in Geometry at Work, C.A. Gorini, ed., MAA Notes 53},

year = {},

pages = {174--194},

publisher = {America}

}

### OpenURL

### Abstract

Graph theory owes many powerful ideas and constructions to geometry. Several well-known families of graphs arise as intersection graphs of certain geometric objects. Skeleta of polyhedra are natural sources of graphs. Operations on polyhedra and maps give rise to various interesting graphs. Another source of graphs are geometric configurations where the relation of incidence determines the adjacency in the graph. Interesting graphs possess some inner structure which allows them to be described by labeling smaller graphs. The notion of covering graphs is explored.

### Citations

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Citation Context ...matrix, and permutation, but we do not require any advanced knowledge of any of these topics. We do not give any rigorous definition of surface or map on a surface. Books listed among our references, =-=[5, 6, 7, 8, 11, 13, 14, 15, 18, 28, 29, 31, 32, 35, 37, 59, 66, 67, 68]-=- provide a spectrum of bacground material spanned Work supported in part by the grants J1-6161 and J26193 of Ministry of Science and Technology of Slovenia. between motivating and introductory chapter... |

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Citation Context ...matrix, and permutation, but we do not require any advanced knowledge of any of these topics. We do not give any rigorous definition of surface or map on a surface. Books listed among our references, =-=[5, 6, 7, 8, 11, 13, 14, 15, 18, 28, 29, 31, 32, 35, 37, 59, 66, 67, 68]-=- provide a spectrum of bacground material spanned Work supported in part by the grants J1-6161 and J26193 of Ministry of Science and Technology of Slovenia. between motivating and introductory chapter... |

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Introduction to Graph Theory
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Citation Context ...matrix, and permutation, but we do not require any advanced knowledge of any of these topics. We do not give any rigorous definition of surface or map on a surface. Books listed among our references, =-=[5, 6, 7, 8, 11, 13, 14, 15, 18, 28, 29, 31, 32, 35, 37, 59, 66, 67, 68]-=- provide a spectrum of bacground material spanned Work supported in part by the grants J1-6161 and J26193 of Ministry of Science and Technology of Slovenia. between motivating and introductory chapter... |

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Citation Context ...ces results in a connected subgraph. It is planar if it can be drawn in the plane without crossings. Let us call a graph polyhedral if it is a one-skeleton of a convex polyhedron. Theorem 3 (Steinitz =-=[63]-=-.) The one-skeleton of an arbitrary convex polyhedron is a planar 3connected graph and each planar 3-connected graph is polyhedral. 4 Geometry at Work in Mathematics and Science Figure 3: Two alternat... |

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Citation Context ...nly if it has no chordless cycle Cn , for n ? 3 and its complement admits a transitive orientation. Unit sphere graphs in 1D are called unit interval graphs. They were characterized by F. S. Roberts, =-=[60]-=-. 1 In chemistry a benzenoid graph is sometimes defined in a slightly different way. Namely, it is required to have at least one Kekule structure, i.e. a perfect matching. Also, a long string of hexag... |

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Citation Context ...e the 0-th line. The i-th line is then obtained by adding i (mod n) to the index of each element of the 0-th line. The Levi graph L = L(P; B) of a configuration was introduced by Coxeter in 1950, see =-=[12]. It is a bipartite -=-graph with "black" vertices P and "white" vertices B and with an edge between p 2 P and B 2 B if and only if p 2 B. Note that dual configurations have the same Levi graph but the r... |

35 |
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30 |
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Citation Context ...h of girth 10 on 62 or less vertices. Since the 9-cage has 58 vertices and the 10-cage has 70 vertices the respective gaps are 12 for the 9-cage and only 8 for the 10-cage. For a survey on cages, see =-=[69]-=-. There is a way of describing certain large graphs using labels on smaller ones. We will introduce this method using the cages as examples. Let us start with the 5-cage, the Petersen graph. 18 Geomet... |

22 |
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17 |
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Citation Context ... r) has vertex set fu 0 ; u 1 ; : : : ; un\Gamma1 ; v 0 ; v 1 ; : : : ; vn\Gamma1 g and edges of the form u i v i ; u i u i+1 ; v i v i+r ; i 2 f0; 1; : : : ; n \Gamma 1g with arithmetic modulo n. In =-=[21]-=- the automorphism group of GP (n; r) was determined for each n and r. With the exception of the dodecahedron GP (10; 2), the generalized Petersen graph GP (n; r) is vertex transitive, if and only if r... |

15 |
Regular polytopes (Third edition
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15 |
Zero-symmetric graphs
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14 |
Diagonal flips of triangulations on surfaces
- Negami
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(Show Context)
Citation Context ...ation. Note that introduction of any additional edge to such a graph necessarily produces a non-planar graph, i.e., a graph that cannot be drawn in the plane without crossing of lines. Diagonal flips =-=[52]-=- It can be shown that any two triangulations on the same number of vertices are equivalent under diagonal flips: this means that one can start with one triangulation and obtain any other by a series o... |

11 |
Finite Geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete
- Dembowski
- 1968
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Citation Context |

9 | The combinatorial distance geometry approach to the calculation of molecular conformation - Havel, Kuntz, et al. - 1983 |

8 | Fast generation of cubic graphs - Brinkmann |

8 | Leapfrog Transformation and polyhedra of Clar Type
- PW, Pisanski
- 1994
(Show Context)
Citation Context ...(G) = PSi(Me(Su1(G))). It is the medial of a 1-subdivided map. That would introduce parallel edges and digons. That is why we insist that the operation is followed by parallel simplification. Compare =-=[20]-=-. ffl Su2: 2-dimensional subdivision The 2-dimensional subdivision of a graph is obtained by adding a vertex in the center of each face and joining it by edges to the vertices of the original face; se... |

5 | Space graphs and sphericity - Maehara - 1984 |

5 | Erzeugung regulärer Graphen - Meringer - 1996 |

5 | Symmetry Properties of Graphs of Interest in Chemistry 11 - RandiC - 1979 |

4 |
The smallest cubic graphs of girth 9
- Brinkmann, McKay, et al.
- 1995
(Show Context)
Citation Context ...sult came as a surprise. There are 18 non-isomorphic 9cages. All smaller cages have regular structure and are all unique. However, the 9-cages do not show any apparent structure; they are computed in =-=[10]-=-. Balaban found one of the three 10-cages which is shown in Figure 28. It is perhaps of interest to note that the 10-cages were known before all the 9cages were computed. The reason is simply in the f... |

4 |
Configurations and graphs
- Gropp
- 1993
(Show Context)
Citation Context ... configuration with no triangles [66]; see Figure 23. For a more thorough introduction to the interesting area of configurations the reader is referred to the work of Harald Gropp; see, for instance, =-=[24]-=-, [23], [26], [25]. Algorithmic aspects are covered in [6] and [64]. Configurations used to play important role in geometry. For instance, the following Theorem is one of the main results of a PhD the... |

4 |
The remarkable generalized Petersen graph G(8
- Marusic, Pisanski
(Show Context)
Citation Context ...3), the only other generalized Petersen graph that is a Haar graph, is isomorphic to H(133). Its automorphism group is of interest in topological graph theory; see [65]. We specialize the result from =-=[45]-=- to Levi graphs and combine it with common knowledge. Proposition 1 ([45]) GP (8; 3) is the only generalized Petersen graph that is a Levi graph of a cyclic configuration. The other two generalized Pe... |

4 |
There is only one group of genus two
- Tucker
- 1984
(Show Context)
Citation Context ... 1) = H(2 2m\Gamma1 + 3) and GP (8; 3), the only other generalized Petersen graph that is a Haar graph, is isomorphic to H(133). Its automorphism group is of interest in topological graph theory; see =-=[65]-=-. We specialize the result from [45] to Levi graphs and combine it with common knowledge. Proposition 1 ([45]) GP (8; 3) is the only generalized Petersen graph that is a Levi graph of a cyclic configu... |

3 |
cycles in cubic cayley graphs on dihedral groups
- Hamilton
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(Show Context)
Citation Context ...t more than one curved line is needed; see Figure 18. A modern version of Theorem 5 can be found in algorithmic form in [6] and has also been implemented as a computer program [57]. Alspach and Zhang =-=[1]-=- proved that every cubic Cayley graph of a dihedral group is Hamiltonian. Compare also [2]. This result covers also cubic Haar graphs. Hence it applies to cyclic configurations. Proposition 2 (Alspach... |

3 |
Schur Norms of Haar Graphs
- Hladnik, Pisanski
(Show Context)
Citation Context ...information about a cyclic configuration and its Levi graph can be encoded by a positive integer N . In this way we get a graph H(N) for each integer N which is called the Haar graph of N ; see [34], =-=[33]-=-. If its girth is at least 6 then H(N) is a Levi graph of a cyclic configuration. In order to make the definition clear, let us construct the Heawood graph as the Haar graph H(69). Since 69 is written... |

3 | Graph-drawing Algorithms Geometries Versus Molecular Mechanics in Fullerenes - Kaufman, Pisanski, et al. - 1996 |

3 | Petkovsek: Intersection graphs of halflines and halfplanes - Klavzar, M - 1987 |

3 |
Graph geometry, graph metrics
- Klein
- 1997
(Show Context)
Citation Context ...n graph theory it is not widely known that a graph can allow additional metrics. Besides the common shortest path metric we also have the resistance distance metric and other metric alternatives; see =-=[40]-=- and [41]. 3 Polyhedra and Graphs In the previous section we found a way from a polyhedron to a graph via unit sphere graphs. There is a much easier way of getting a graph from a polyhedron, obtained ... |

3 |
A note on the generalized Petersen graphs that are also Cayley graphs
- Lovrecic-Sarazin
- 1997
(Show Context)
Citation Context ...(n; r) was determined for each n and r. With the exception of the dodecahedron GP (10; 2), the generalized Petersen graph GP (n; r) is vertex transitive, if and only if r 2 j \Sigma1(mod n); see also =-=[43]-=-. It was also shown in [21] that GP (n; r) is arc-transitive if and only if (n; r) 2 f(4; 1); (5; 2); (8; 3); (10; 2); (10; 3); (12; 5); (24; 5)g: Note that GP (4; 1) is the cube Q 3 . On the other ha... |

3 | Uniform Polyhedra
- Maeder
- 1993
(Show Context)
Citation Context ...yhedra giving the same one-skeleton. The uniform polyhedron of Figure 4 is a model of a projective plane. In [66] it is called a heptahedron. However, it is better known as a tetrahemihexahedron; see =-=[49]-=-, where you can find more information about and illustrations of uniform polyhedra. It has 6 vertices, 12 edges, 4 triangles and 3 squares. Its one-skeleton is again K 2;2;2 ; see Figure 3. Figure 5 s... |

2 |
On a class of Hamilton laceable 3-regular graphs
- Alspach, Chen, et al.
- 1996
(Show Context)
Citation Context ... found in algorithmic form in [6] and has also been implemented as a computer program [57]. Alspach and Zhang [1] proved that every cubic Cayley graph of a dihedral group is Hamiltonian. Compare also =-=[2]-=-. This result covers also cubic Haar graphs. Hence it applies to cyclic configurations. Proposition 2 (Alspach and Zhang) A Levi graph of a cyclic n 3 configuration is Hamiltonian. One can show that a... |

2 | Regular Linear Spaces. Beitrage zur Algebra und Geometrie - Betten, Betten - 1997 |

2 |
Counting symmetric configurations v 3 , submitted
- Betten, Brinkmann, et al.
(Show Context)
Citation Context ...ycle in G, the girth, is at least 6. Table 17 shows the correspondence between properties of configurations and their Levi graphs. Recently the number of n 3 configurations up to ns18 was computed in =-=[4]-=-. Example 5 Each regular graph of valence k without loops and multiple edges can be viewed as an (n k ; b 2 ) configuration of points and lines. Vertices correspond to points and edges correspond to l... |

2 | On 2-arc-transitive covers of complete graphs, submitted - Du, Marusic, et al. |

2 |
On the history of configurations
- Gropp
- 1990
(Show Context)
Citation Context ...guration with no triangles [66]; see Figure 23. For a more thorough introduction to the interesting area of configurations the reader is referred to the work of Harald Gropp; see, for instance, [24], =-=[23]-=-, [26], [25]. Algorithmic aspects are covered in [6] and [64]. Configurations used to play important role in geometry. For instance, the following Theorem is one of the main results of a PhD thesis of... |

2 |
Configurations and their realizations
- Gropp
- 1997
(Show Context)
Citation Context ...on with no triangles [66]; see Figure 23. For a more thorough introduction to the interesting area of configurations the reader is referred to the work of Harald Gropp; see, for instance, [24], [23], =-=[26]-=-, [25]. Algorithmic aspects are covered in [6] and [64]. Configurations used to play important role in geometry. For instance, the following Theorem is one of the main results of a PhD thesis of renow... |

2 |
Groups and their graphs. Random House
- Grossman, Magnus
- 1966
(Show Context)
Citation Context |

2 |
Multilayered cyclic fence graphs: novel cubic graphs related to the graphite network
- Hosoya, Obnna, et al.
- 1995
(Show Context)
Citation Context ...thermore, vertex v 0 is adjacent Bridges between Geometry and Graph Theory 15 Figure 20: Four views of the Mobius-Kantor graph GP (8; 3) = H(133). A modification of the second drawing can be found in =-=[36]-=-. to vertices u 0 ; u 4 ; u 6 and vertex v i is adjacent to the vertices u0+i ; u4+i ; u6+i , where addition is taken mod 7. Note that we constructed in passing the Fano configuration: ff0; 4; 6g; f1;... |

2 | All-Conjugated Carbon Species - Klein, Zhu - 1997 |

2 | NiceGraph Program and its applications in chemistry, Croat - Pisanski, Plestenjak, et al. - 1995 |

2 | Generating Fullerenes at Random - Plestenjak, Pisanski, et al. - 1996 |

2 |
Rotagraphs and their generalizations
- Pisanski, Zitnik, et al.
- 1994
(Show Context)
Citation Context ...e labels: the trivial 0 and the non-trivial 1. In this case the covering graph is called a rotagraph. The subgraph M of G composed of those edges of G with trivial voltages is called a monograph; see =-=[56]-=- for more information on rotagraphs. Many well-known graphs can be described as rotagraphs, for example the prisms and antiprisms. However, the Petersen graph is not a rotagraph although it has a rota... |

1 |
Bouwer et al, editors. The Foster Census. The Charles Babbage Research
- Z
- 1988
(Show Context)
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1 |
von Sterneck. Die Configurationen 11 3
- Daublebsky
(Show Context)
Citation Context ...h is also transitive and self-dual; see Figure 22. There are 31 configurations of type 11 3 and 229 of type 12 3 . These facts were established more than hundred years ago by Daublebsky von Sterneck, =-=[16]-=-, [17], although he missed one 12 3 configuration. Only recently it has been shown [64] that all these configurations admit realizations in 3D using integer coordinates. The Cremona-Richmond configura... |

1 |
The Construction of All Configurations (12 4 ; 16 3
- Gropp
- 1992
(Show Context)
Citation Context ...h no triangles [66]; see Figure 23. For a more thorough introduction to the interesting area of configurations the reader is referred to the work of Harald Gropp; see, for instance, [24], [23], [26], =-=[25]-=-. Algorithmic aspects are covered in [6] and [64]. Configurations used to play important role in geometry. For instance, the following Theorem is one of the main results of a PhD thesis of renowned Ge... |