On the weights of simple paths in weighted complete graphs

TL;DR

This paper develops criteria to determine when a given set of path weights can be realized by a weighted complete graph, addressing the inverse problem of graph weight reconstruction.

Contribution

It introduces new conditions for the existence of a weighted complete graph matching specified path weight multisets, advancing understanding of graph weight configurations.

Findings

Established a criterion for when a multisubset corresponds to a path weight set between two vertices.

Provided conditions for the existence of a complete graph matching given multisets for all vertex pairs.

Extended the criteria to the case of all pairs simultaneously.

Abstract

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