Realization of distance matrices by graphs of genus 1

TL;DR

This paper investigates the conditions under which a given distance matrix can be realized by graphs of genus 1, providing an algorithm for recognition and construction of such graphs.

Contribution

It introduces a novel algorithm to determine and construct realizations of distance matrices by genus 1 graphs, focusing on practical applications.

Findings

Algorithm successfully identifies realizable matrices

Constructs explicit genus 1 graph realizations

Provides theoretical insights into distance matrix realization

Abstract

Given a distance matrix $D$, we study the behavior of its compaction vector and reduction matrix with respect to the problem of the realization of $D$ by a weighted graph. To this end, we first give a general result on realization by $n-$cycles and successively we mainly focus on graphs of genus 1, presenting an algorithm which determines when a distance matrix is realizable by such a kind of graph, and then, shows how to construct it.