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PreprintCC BY 4.0arXiv

On the Graph Corresponding to Center of Distances of Metric Space

  • Oleksiy Dovgoshey
  • Olga Rovenska
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Abstract

The center of distances C(Y) of a metric space (Y,ρ) is the set of all t≥ 0 for which the equation $ρ(y,p)=t $ has a solution for each point p∈ Y. We say that a simple graph CGY is the central graph of a metric space (Y,ρ) if Y is the vertex set of CGY and distinct vertices x,y∈ Y are adjacent if and only if ρ(x,y)∈ C(Y). We prove the inequality $|E(CG_Y)| \geq \left\lceil \frac{1}{2}|Y| \right\rceil $ for all finite metric spaces (Y,ρ) with |Y|≥ 2 and non-empty CGY. It is also proved that the inequality $|E(CG_X)| \geq |X|-1 $ holds for each finite ultrametric space (X,d). The central graphs of metric spaces (Y,ρ) and ultrametric spaces (X,d) satisfying $|E(CG_Y)|=\left\lceil \frac{1}{2}|Y| \right\rceil $ and, respectively, $|E(CG_X)|=|X|-1 $ are described up to graph isomorphism.

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Cite

BibTeXRIS
BibTeX
@misc{dovgoshey2026graph,
  title = {{On the Graph Corresponding to Center of Distances of Metric Space}},
  author = {Oleksiy Dovgoshey and Olga Rovenska},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11596},
  eprint = {2610.11596},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11596},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the Graph Corresponding to Center of Distances of Metric Space
AU  - Oleksiy Dovgoshey
AU  - Olga Rovenska
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11596
UR  - https://arxiv.org/abs/2610.11596
AB  - The center of distances C(Y) of a metric space (Y,ρ) is the set of all t≥ 0 for which the equation $ρ(y,p)=t $ has a solution for each point p∈ Y. We say that a simple graph CG_Y is the central graph of a metric space (Y,ρ) if Y is the vertex set of CG_Y and distinct vertices x,y∈ Y are adjacent if and only if ρ(x,y)∈ C(Y). We prove the inequality $|E(CG_Y)| \geq \left\lceil \frac{1}{2}|Y| \right\rceil $ for all finite metric spaces (Y,ρ) with |Y|≥ 2 and non-empty CG_Y. It is also proved that the inequality $|E(CG_X)| \geq |X|-1 $ holds for each finite ultrametric space (X,d). The central graphs of metric spaces (Y,ρ) and ultrametric spaces (X,d) satisfying $|E(CG_Y)|=\left\lceil \frac{1}{2}|Y| \right\rceil $ and, respectively, $|E(CG_X)|=|X|-1 $ are described up to graph isomorphism.
ER  -