Name
Abstract | ||
|---|---|---|
The problem of finding multiloop networks with a fixed number of vertices and small diameter has been widely studied. In this work, we study the triple loop case of the problem by using a geometrical approach which has been already used in the double loop case. Given a fixed number of vertices N , the general problem is to find ‘steps’ s 1 , s 2 , …, s d ∈ Z N , such that the diagraph G ( N ; s 1 , s 2 , …, s d ), with set of vertices V = Z N and adjacencies given by v → v + s i ( mod N ), i = 1, 2, …, d , has minimum diameter D ( N ). A related problem is to maximize the number of vertices N ( d , D ) when the degree d and the diameter D are given. In the double loop case ( d = 2) it is known that N(2, D) = ⌈ 1 3 (D+2) 2 ⌉ − 1 . Here, a method based on lattice theory and integral circulant matrices is developed to deal with the triple loop case ( d = 3). This method is then applied for constructing three infinite families of triple loop networks with large order for the values of the diameter D ≡ 2, 4, 5( mod 6), showing that N(3, D) ⩾ 2 27 D 3 + O(D 2 ) . Similar results are also obtained in the more general framework of (triple) commutative-step digraphs. |
Year | DOI | Venue |
|---|---|---|
1997 | 10.1016/S0012-365X(96)00213-0 | Discrete Mathematics |
Keywords | Field | DocType |
small transmission delay,triple loop network,lattice theory,circulant matrices | Discrete mathematics,Monad (category theory),Combinatorics,Vertex (geometry),Lattice (order),Transmission delay,Circulant matrix,Mathematics | Journal |
Volume | ISSN | Citations |
167-168, | Discrete Mathematics | 10 |
PageRank | References | Authors |
1.03 | 8 | 3 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| F. Aguiló | 1 | 56 | 5.74 |
| M. A. Fiol | 2 | 816 | 87.28 |
| C. Garcia | 3 | 10 | 1.37 |