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Symmetric and semisymmetric graphs construction using G-graphs

Published: 24 July 2005 Publication History

Abstract

Symmetric and semisymmetric graphs are used in many scientific domains, especially parallel computation and interconnection networks. The industry and the research world make a huge usage of such graphs. Constructing symmetric and semisymmetric graphs is a large and hard problem. In this paper a tool called G-graphs and based on group theory is used. We show the efficiency of this tool for constructing symmetric and semisymmetric graphs and we exhibit experimental results.

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    Robert Bruce King

    A graph that is both edge-transitive and vertex-transitive is called a symmetric graph. A graph that is edge-transitive, but not vertex-transitive is called a semisymmetric graph. Constructing symmetric and semisymmetric graphs is a large and hard problem. This paper presents an algorithm based on group theory to construct symmetric and semisymmetric graphs. Let (G, S) be a group with a set of generators S = {s1, s2, and sk}. For any s __ __ S, consider the resulting cycles (s)x = (x, sx, s2x, so(s)-1x) of the permutation gs: x __ __ Lsx, where o(s) is the order of the element s. Now define a new graph Φ (G; S) as follows: (1) The vertices of Φ (G; S) are the cycles of gs, s __ __ S. (2) For all pairs of cycles (s)x, (t)y, {(s)x, (t)y} is an edge of multiplicity p if card((s)x n (t)y) = p and p = 1. Thus Φ (G; S) is a k-partite graph, and any vertex has an o(s) loop. Denote as = (G; S) the graph F(G; S) with the loops removed. The graphs Φ (G; S) and = (G; S) are called graphs from a group or, more specifically, G-graphs generated by the group (G, S). Using this terminology, the following theorems are proved in this paper: Proposition 1: Let Φ (G; S) be a G-graph. This graph is connected if and only if S is a generator set of G. Proposition 2: Let h be a morphism between (G 1, S 1) and (G 2, S 2). Then, there exists a morphism Φ(h) between Φ (G 1; S 1) and Φ (G 2; S 2). Theorem 1: Let G 1 and G 2 be two Abelian groups. These two groups are isomorphic if and only if Φ (G 1; S 1) and Φ (G 2, S 2) are isomorphic. Proposition 3: Let Γ be a connected bipartite and regular G-graph of degree p, where p is a prime number. Then, either Γ is simple, or Γ is of order 2. The key theorem in this paper is then Theorem 3: Let Γ be a bipartite connected semiregular simple graph with vertices divided into two sets V1 and V2. Let (G, {s1, s2}) be a group with o (s1) = deg(x), x be a vertex in set V1, and o(s2) = deg(y), y be a vertex in set V2. The following properties are then equivalent: (i) The graph Γ is a G-graph, = (G; {s 1, s 2}). (ii) The line graph L (Γ) is a Cayley graph Cay(H; A) where A = a 1a 2 \ {e} with (G, {s 1, s 2}) ≂ (H, {a 1, a 2}) (iii) The group G is a subgroup of the automorphism group of Γ, which acts regularly on the set of edges of Γ. Using this theorem, an algorithm was developed for the construction of symmetric and semisymmetric graphs. Tables of quartic symmetric simple G-graphs, quartic semisymmetric simple G-graphs, cubic semisymmetric G-graphs, quintic semisymmetric G-graphs, and quintic symmetric G-graphs are presented in the paper. Online Computing Reviews Service

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    cover image ACM Conferences
    ISSAC '05: Proceedings of the 2005 international symposium on Symbolic and algebraic computation
    July 2005
    388 pages
    ISBN:1595930957
    DOI:10.1145/1073884
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    Published: 24 July 2005

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    Author Tags

    1. G-graphs
    2. graphs from group
    3. semisymmetric graph
    4. symmetric graphs

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    • (2010)𝔾-planar abelian groupsInvolve, a Journal of Mathematics10.2140/involve.2010.3.2333:2(233-240)Online publication date: 11-Aug-2010
    • (2008)G-Graphs and Algebraic HypergraphsElectronic Notes in Discrete Mathematics10.1016/j.endm.2008.01.02730(153-158)Online publication date: Feb-2008
    • (2008) -graphs: An efficient tool for constructing symmetric and semisymmetric graphs Discrete Applied Mathematics10.1016/j.dam.2007.11.011156:14(2719-2739)Online publication date: Jul-2008
    • (2007)G-graphs for the cage problemProceedings of the 2007 international symposium on Symbolic and algebraic computation10.1145/1277548.1277556(49-53)Online publication date: 29-Jul-2007
    • (2007)A New Property of Hamming Graphs and Mesh of d-ary TreesComputer Mathematics10.1007/978-3-540-87827-8_11(139-150)Online publication date: 15-Dec-2007
    • (2005)Symmetry and connectivity in G-graphsElectronic Notes in Discrete Mathematics10.1016/j.endm.2005.06.09622(481-486)Online publication date: Oct-2005

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