Rubei, Prof. Elena (2012) On the weights of simple paths in weighted complete graphs. Technical Report 2012.01.02, no. 1, Dipartimento di Matematica "U.Dini", Universita' di Firenze.

Abstract

Consider a weighted graph G with n vertices, numbered by the set {1, ..., n}. For any path p in G, we call wG(p) the sum of the weights of the edges of the path and we define the multiset Di,j(G) = {wG(p)| p simple path between iand j} We establish a criterion to say when, given a multisubset of R, there exists a weighted complete graph G such that the multisubset is equal to Di,j (G) for some i, j vertices of G. Besides we establish a criterion to say when, given for any i, j ∈ {1, ..., n} a multisubset of R, Di,j , there exists a weighted complete graph G with vertices {1, ..., n} such that Di,j (G) = Di,j for any i, j.