The forwarding indices of graphs - a survey

Abstract

Abstract. A routing R of a connected graph G of order n is a collection of n(n − 1) simple paths connecting every ordered pair of vertices of G. The vertex-forwarding index ξ(G, R) of G with respect to a routing R is defined as the maximum number of paths in R passing through any vertex of G. The vertex-forwarding index ξ(G) of G is defined as the minimum ξ(G, R) over all routings R of G. Similarly, the edge-forwarding index π(G, R) of G with respect to a routing R is the maximum number of paths in R passing through any edge of G. The edge-forwarding index π(G) of G is the minimum π(G, R) over all routings R of G. The vertex-forwarding index or the edge-forwarding index corresponds to the maximum load of the graph. Therefore, it is important to find routings minimizing these indices and thus has received much research attention for over twenty years. This paper surveys some known results on these forwarding indices, further research problems and several conjectures, also states some difficulty and relations to other topics in graph theory.

How to Design Graphs with Low Forwarding Index and Limited Number of Edges

“…To our surprise, we could not find the following result in the literature, moreover in the recent survey [11] the best bounds for cubic graphs were provided by shuffle exchange graphs, and more generally, for bounded degree graphs the best bounds known are derived using de Bruijn graphs. Those bounds are rather good since they differ from the lower bound only by a relatively small (always lesser than 2) constant factor.…”

“…For ∆ = 3, when e = 3n 2 , graphs such like the shuffle exchange provide deterministic generic constructions for which π(G) ≤ n log 2 n (this is a folk result for people studying network throughput, one may see [26]). Since using the Moore bound (that bound claims by direct counting that the average distance in a ∆ bounded degree graph is of order log ∆−1 (n), see as example [16]) one can prove that π * (n, 3n…”

“…Moreover for larger values of ∆ de Bruijn graphs and their variants provide ∆-regular graphs whose forwarding index is of the right order (see Figure 2). So when the degree is bounded by ∆, the value of π(n, ∆ 2 n) is relatively well understood (see [6,11]), and structures close to the optimal are obtained using de Bruijn graphs or slight variants of it. Indeed, on the one hand, the Moore bound implies that :…”

“…This notion has also been used to prove that some Markov chains mix fast using either canonical paths (routings) or "resistance" [21]. See the recent survey [26] for a global view on the known results. The problem is also known as the maximum concurrent flow problem and its dual was probably first introduced in [20] in which the authors also discussed the relation with the network throughput, in [23] a simple oblivous packet routing algorithm achieving network stability for any rate λ with λπ < 1 was provided.…”

Forwarding and optical indices of 4-regular circulant networks

“…If s = √ n, then q = ⌊n/s⌋ = √ n and by (27) we obtain − → π (C n (1, s)) ≤ ( √ n(n − ǫ(s))/8. If √ n + 1 ≤ s < n/2, then by (28) and (29) we have − → π (C n (1, s)) ≤ (s(n + r + 2) − ǫ(s)q)/8 = (s 2 (n + r + 2) − ǫ(s)(n − r))/8s.…”