Line digraph iterations and the (d,k) problem for directed graphs

Abstract

We consider in this paper the (d,k) problem for directed graphs: to maximize the number of vertices in a digraph of degree d and diameter k. For any values of d and k, we construct a graph with a number of vertices larger than (d 2-1)/d 2 times the (non-attainable) Moore bound. In particular, this solves the (d,k) digraph problem for k=2. We also show that these graphs can be obtained as line digraph iterations and that this technique provides us with a simple local routing algorithm for the corresponding networks.

Iterated line digraphs are asymptotically dense

“…Moreover, if G is a digraph (different from a directed cycle) with diameter k and maximum eigenvalue λ 0 , then its -iterated line digraph L (G) has diameter k = k + (see Fiol, Yebra, and Alegre [3,4]), maximum eigenvalue λ 0 (the line digraph technique preserves all the eigenvalues, see Balbuena, Ferrero, Marcote, and Pelayo [1]), and number of vertices…”

On Moore bipartite digraphs

“…We point out that, from the results in [7,8], if G is a Moore bipartite digraph with diameter k ¼ 3 and out-degrees…”

“…Thus, in Section 2, we reformulate such a search in polynomial and additive terms, which allows us to construct new Moore bipartite digraphs of diameter three and composite out-degrees by using the so-called De Bruijn nearfactorizations of cyclic groups. From such optimal digraphs, and using the iterated line digraph technique introduced in [7,8], we can obtain new families of dense bipartite digraphs with arbitrary diameter.…”

Nonexistence of Almost Moore Digraphs of Degrees 4 and 5 with Self-Repeats

“…Concerning the existence of such (d, k)-digraphs, Fiol et al showed in [12] that (d, 2)digraphs do exist for any degree d > 1 and Gimbert completed their classification for k = 2 in [14]. But so far, it seems that they do not exist for the remaining values of the diameter.…”

“…have all vertices with out-degree d and its diameter must be k (see [12]). Moreover, its in-degrees are also d (see [17]).…”