The groups of the generalized Petersen graphs
01 Sep 1971-Vol. 70, Iss: 2, pp 211-218
TL;DR: In this paper, the generalized Petersen graph G(n, k) was defined for integers n and k with 2 ≤ 2k < n, and G(5, 2) is the well known Petersen graph.
Abstract: 1. Introduction. For integers n and k with 2 ≤ 2k < n, the generalized Petersen graph G(n, k) has been defined in (8) to have vertex-setand edge-set E(G(n, k)) to consist of all edges of the formwhere i is an integer. All subscripts in this paper are to be read modulo n, where the particular value of n will be clear from the context. Thus G(n, k) is always a trivalent graph of order 2n, and G(5, 2) is the well known Petersen graph. (The subclass of these graphs with n and k relatively prime was first considered by Coxeter ((2), p. 417ff.).)
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TL;DR: In this article, it was shown that if p is a prime, k and m =< P are positive integers, and r is a vertex symmetric digraph of order p^k or m^p, then r has an automorphism all of whose orbits have cardinality p.
124 citations
TL;DR: In this paper, the problem of determining the order of a vertex-transitive graph which is not a Cayley graph was studied, and a sequence of constructions which solved the problem for many orders was presented.
Abstract: The Petersen graph on 10 vertices is the smallest example of a vertex-transitive graph which is not a Cayley graph. We consider the problem of determining the orders of such graphs. In this, the first of a series of papers, we present a sequence of constructions which solve the problem for many orders. In particular, such graphs exist for all orders divisible by a fourth power, and all even orders which are divisible by a square.
82 citations
Additional excerpts
...A non-Cayley vertex-transitive graph of order 2p, p ≡ 1 (mod 4), was constructed in [4]....
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TL;DR: It is shown that the graphs for which the Szeged index equals @?G@?@?|G|^24 are precisely connected, bipartite, distance-balanced graphs, which enables us to disprove a conjecture proposed in M.H. Khalifeh, H. Yousefi-Azari, and A.R. Wagner.
Abstract: It is shown that the graphs for which the Szeged index equals @?G@?@?|G|^24 are precisely connected, bipartite, distance-balanced graphs. This enables us to disprove a conjecture proposed in [M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, S.G. Wagner, Some new results on distance-based graph invariants, European J. Combin. 30 (2009) 1149-1163]. Infinite families of counterexamples are based on the Handa graph, the Folkman graph, and the Cartesian product of graph. Infinite families of distance-balanced, non-regular graphs that are prime with respect to the Cartesian product are also constructed.
71 citations
TL;DR: The well-known Petersen graph G(5,2) admits a semi-regular automorphism @a acting on the vertex set with two orbits of equal size that makes it a bicirculant, and it is shown that trivalent bICirculants fall into four classes.
68 citations
Cites background from "The groups of the generalized Peter..."
...The class of vertex-transitive generalized Petersen graphs was established in [10]....
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TL;DR: It is proved that the generalized Petersen graph GP (n, k), n ≤ 3, 1 ≥ k < n/2 is a cubic graph with vertex-set {uj; i ϵ Zn} ∪ {vj; iπ Zn}, and edge-set{uiui, uivi, vivi+k, iϵZn}.
Abstract: The generalized Petersen graph GP (n, k), n ≤ 3, 1 ≥ k < n/2 is a cubic graph with vertex-set {uj; i ϵ Zn} ∪ {vj; i ϵ Zn}, and edge-set {uiui, uivi, vivi+k, iϵZn}. In the paper we prove that
(i) GP(n, k) is a Cayley graph if and only if k2 1 (mod n); and
(ii) GP(n, k) is a vertex-transitive graph that is not a Cayley graph if and only if k2 -1 (mod n) or (n, k) = (10, 2), the exceptional graph being isomorphic to the 1-skeleton of the dodecahedon.
The proof of (i) is based on the classification of orientable regular embeddings of the n-dipole, the graph consisting of two vertices and n parallel edges, while (ii) follows immediately from (i) and a result of R. Frucht, J.E. Graver, and M.E. Watkins [“The Groups of the Generalized Petersen Graphs,” Proceedings of the Cambridge Philosophical Society, Vol. 70 (1971), pp. 211-218]. © 1995 John Wiley & Sons, Inc.
64 citations
References
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Book•
01 Jan 1957
TL;DR: In this article, the authors present a systematic enumeration of cosets based on the following groups: cyclic, Dicyclic and Metacyclic groups, Graphs, Maps, Cayley Diagrams, Hyperbolic Tessellations and Fundamental Groups.
Abstract: 1. Cyclic, Dicyclic and Metacyclic Groups.- 2. Systematic Enumeration of Cosets.- 3. Graphs, Maps and Cayley Diagrams.- 4. Abstract Crystallography.- 5. Hyperbolic Tessellations and Fundamental Groups.- 6. The Symmetric, Alternating, and other Special Groups.- 7. Modular and Linear Fractional Groups.- 8. Regular Maps.- 9. Groups Generated by Reflections.- Tables 1-12.
1,837 citations
Book•
15 Dec 1966
TL;DR: In this paper, a simple path is defined as a path that does not repeat any edges and is non-simple in the sense that it can be non-repeatable from x to y.
Abstract: a b x u y w v c d Typical question: Is it possible to get from some node u to another node v ? Example: Train network – if there is path from u to v , possible to take train from u to v and vice versa. If it's possible to get from u to v , we say u and v are connected and there is a path between u and v Paths a b x u y w v c d A path between u and v is a sequence of edges that starts at vertex u, moves along adjacent edges, and ends in v. A simple path is a path that does not repeat any edges What are all the simple paths from z to w ? What are all the simple paths from x to y? How many paths (can be non-simple) are there from x to y? Connectedness a b x u y w v c d A graph is connected if there is a path between every pair of vertices in the graph Example: This graph not connected; e.g., no path from x to d A connected component of a graph G is a maximal connected subgraph of G Example Prove: Suppose graph G has exactly two vertices of odd degree, say u and v. Then G contains a path from u to v .
636 citations
01 Oct 1947
TL;DR: A cubical graph is a simplicial 1-complex in which each 0-simplex is incident with just three 1-simples as discussed by the authors, and cubical graphs are simplicial simplicial 2-complexes.
Abstract: We begin with some definitions.A cubical graph is a simplicial 1-complex in which each 0-simplex is incident with just three 1-simplexes.
497 citations
337 citations