Weighted graphs with distances in given ranges
Elena Rubei

TL;DR
This paper investigates conditions under which weighted graphs and trees can be constructed so that the shortest path distances between specified vertex pairs fall within given ranges, extending classical distance realization problems.
Contribution
It provides new criteria for the existence of weighted graphs and trees with prescribed distance intervals between vertex pairs, covering both positive and general weights.
Findings
Derived necessary and sufficient conditions for graphs with distance ranges.
Extended results to trees with positive and arbitrary weights.
Established frameworks for constructing such graphs and trees.
Abstract
Let be a weighted simple finite connected graph, that is, let be a simple finite connected graph endowed with a function from the set of the edges of to the set of real numbers. For any subgraph of , we define to be the sum of the weights of the edges of . For any vertices of , we define to be the minimum of the weights of the simple paths of joining and . The are called -weights of . Let and be two families of positive real numbers parametrized by the -subsets of with for any ; we study when there exist a positive-weighted graph and an -subset of the set of its vertices such that $D_I ({\cal G}) \in [m_I,…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph theory and applications · Limits and Structures in Graph Theory
