Characterizing (quasi-)ultrametric finite spaces in terms of (directed) graphs

Abstract

Let D=(V,A) be a complete directed graph (digraph) with a positive real weight function d:A→{d1,…,dk}⊆R+ such that 0<d1<⋯<dk. For every i∈[k]={1,…,k}, let us set Ai={(u,w)∈A∣d(u,w)≤di} and assume that each subgraph Di=(V,Ai),i∈[k], in the obtained nested family is transitive, that is, (u,w)∈Ai whenever (u,v),(v,w)∈Ai for some v∈V. This assumption implies that the considered weighted digraph (D,d) defines a quasi-ultrametric finite space (QUMFS) and, conversely, each QUMFS is uniquely (up to an isometry) is realized by a nested family of transitive digraphs.These simple observations imply important corollaries. For example, each QUMFS can be realized by a multi-pole flow network. Furthermore, k≤(n2)+n−1=12(n−1)(n+2), where n=|V|, and this upper bound for the number k of pairwise distinct distances is precise. Moreover, we characterize all QUMFSes for which the equality holds.In the symmetric case, d(u,w)=d(w,u), we obtain a canonical representation of an ultrametric finite space (UMFS) together with the well-known bound k≤n−1. Interestingly, due to this representation, a UMFS can be viewed as a positional game structure of k players {1,…,k} such that, in every play, they make moves in a monotone strictly decreasing order.