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.
References
[1]
Fred Annexstein, Marc Baumslag, and Arnold L. Rosenberg. Group action graphs and parallel architectures, SIAM J. Comput, 19:544--569, 1990.]]
Sheldon Akers and Balakrishnan Krishnamurthy. Group graphs as interconnection networks, In Proc. 14th Int. Conf. Fault Tolerant Comput, pages 422--427, 1984.]]
A. Bretto and B. Laget. A new graphical representation of a group. Tenth International Conference on Applications of Computer Algebra.(ACA-2004), Beaumont, USA, 23-25 July 2004. National Science Foundation, (NSF), 2004 25-32, (ISBN: 0-9759946-0-3).]]
Marston Conder and Peter Dobcsanyi, Trivalent symmetric graphs on up to 768 vertices, J. Combinatorial Mathematics and Combinatorial Computing 40 (2002), 41--63.]]
Marston Conder, Aleksander Malnic, Dragan Marusic and Primoz Potocnik, A census of semisymmetric cubic graphs on up to 768 vertices, preprint, March 2004.]]
G. Cooperman and L. Finkelstein and N. Sarawagi. Applications of Cayley Graphs. Appl. Algebra and Error-Correcting Codes. Springer Verlag. Lecture Notes in Computer Sciences, Vol. 508 1991, 367--378.]]
Joseph Lauri, Constructing graphs with several pseudosimilar vertices or edges, Discrete Mathematics, Volume: 267, Issue: 1-3, p. 197--211. June 6, 2003.]]
Brendan D. McKay, Computer Science Department, Australian National University, (1981), Practical graph isomorphism, Congressus Numerantium 30, p. 45--87.]]
Gholaminezhad FAshraf A(2024)A Survey on G-Graph of A Survey on G-Graph of GroupSSRN Electronic Journal10.2139/ssrn.4816064Online publication date: 2024
Gu MChang J(2021)A Note on Super Connectivity of the Bouwer GraphJournal of Interconnection Networks10.1142/S021926592142009321:04Online publication date: 16-Jul-2021
A graph X is said to be G -semisymmetric if it is regular and there exists a subgroup G of A := Aut ( X ) acting transitively on its edge set but not on its vertex set. In the case of G = A , we call X a semisymmetric graph. Let p be a ...
ISSAC '07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Constructing some regular graph with a given girth,a given degree and the fewest possible vertices is a hard problem. This problem is called the cage graph problem and has some links with the error codes theory. In this paper we presents some new graphs,...
A regular edge-transitive graph is said to be semisymmetric if it is not vertex-transitive. By Folkman [J. Folkman, Regular line-symmetric graphs, J. Combin Theory 3 (1967) 215-232], there is no semisymmetric graph of order 2p or 2p^2 for a prime p and ...
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.
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Gholaminezhad FAshraf A(2024)A Survey on G-Graph of A Survey on G-Graph of GroupSSRN Electronic Journal10.2139/ssrn.4816064Online publication date: 2024
Gu MChang J(2021)A Note on Super Connectivity of the Bouwer GraphJournal of Interconnection Networks10.1142/S021926592142009321:04Online publication date: 16-Jul-2021
Bretto AGillibert LWang D(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
Bretto AJaulin CGillibert LLaget B(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
Bretto AGillibert L(2005)Symmetry and connectivity in G-graphsElectronic Notes in Discrete Mathematics10.1016/j.endm.2005.06.09622(481-486)Online publication date: Oct-2005