The Degree-Diameter Problem for Several Varieties of Cayley Graphs I: The Abelian Case

We consider the degree-diameter problem for Cayley graphs of Abelian groups (Abelian graphs) for both directed graphs and undirected graphs. The problem is closely related to that of finding efficient lattice coverings of Euclidean space by shapes such as octahedra and tetrahedra; we exploit this relationship in both directions. For two generators (dimensions), these methods yield optimal Abelian graphs with a given diameter k. (The results in two dimensions are not new; they are given in the literature of distributed loop networks.) We find an undirected Abelian graph with three generators and a given diameter k, which we conjecture to be as large as possible; for the directed case, we obtain partial results. These results are connected to efficient lattice coverings of ${\bf R}^3$ by octahedra or by tetrahedra; computations on Cayley graphs lead us to such lattice coverings, which we conjecture to be optimal. (The problem of finding such optimal coverings can be reduced to a finite number of nonlinear optimization problems.) We discuss the asymptotic behavior of the Abelian degree-diameter problem for large numbers of generators. The graphs obtained here are substantially better than traditional toroidal meshes, but, in the simpler undirected cases, retain certain desirable features such as good routing algorithms, easy constructibility, and the ability to host mesh-connected numerical algorithms without any increase in communication times.

[1]  Jhing-Fa Wang,et al.  Reliable circulant networks with minimum transmission delay , 1985 .

[2]  D. Frank Hsu,et al.  Distributed Loop Computer Networks: A Survey , 1995, J. Parallel Distributed Comput..

[3]  Douglas J. Hoylman.  THE DENSEST LATTICE PACKING OF TETRAHEDRA , 1970 .

[4]  G. Sabidussi.  Vertex-transitive graphs , 1964 .

[5]  Hajime Urakawa.  On the least positive eigenvalue of the Laplacian for the compact quotient of a certain Riemannian symmetric space , 1980 .

[6]  Gérard D. Cohen,et al.  Covering Radius 1985–1994 , 1997, Applicable Algebra in Engineering, Communication and Computing.

[7]  F. Chung.  Diameters and eigenvalues , 1989 .

[8]  Istvan Fary,et al.  Sur la densité des réseaux de domaines convexes , 1950 .

[9]  P. Gritzmann.  Lattice covering of space with symmetric convex bodies , 1985 .

[10]  Fan Chung Graham,et al.  An Upper Bound on the Diameter of a Graph from Eigenvalues Associated with its Laplacian , 1994, SIAM J. Discret. Math..

[11]  Aileen Mary Mcloughlin,et al.  On the covering radius. , 1977 .

[12]  Fred S. Annexstein,et al.  On the diameter and bisector size of Cayley graphs , 1993, Mathematical systems theory.

[13]  Gérard D. Cohen,et al.  Covering radius - Survey and recent results , 1985, IEEE Trans. Inf. Theory.

[14]  Jack Lamoreaux,et al.  Lattice-Simplex Coverings and the 84-Shape , 2000, SIAM J. Discret. Math..

[15]  Sheng Chen,et al.  Undirected loop networks , 1993, Networks.

[16]  C. A. Rogers.  Lattice coverings of space , 1959 .

[17]  R. Dougherty,et al.  Covering radius computations for binary cyclic codes , 1991 .

[18]  R. Stanton,et al.  Note on a “Square” Functional Equation , 1970 .

[19]  Miquel Àngel Fiol Mora,et al.  The Diameter of undirected graphs associated to plane tessellations , 1985 .

[20]  Charles M. Fiduccia,et al.  Geometry and Diameter Bounds of Directed Cayley Graphs of Abelian Groups , 1998, SIAM J. Discret. Math..

[21]  Michael J. Dinneen,et al.  New results for the degree/diameter problem , 1994, Networks.

[22]  Chak-Kuen Wong,et al.  A Combinatorial Problem Related to Multimodule Memory Organizations , 1974, JACM.

[23]  F. Aguiló,et al.  Triple loop networks with small transmission delay , 1997, Discret. Math..