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Classification of strictly resistance nonnegative graphs

  • Hailey Jay Garcia
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Abstract

We say that a graph is resistance nonnegative or RN if it admits a positive edge-weight that yields nonnegative resistance curvature in the sense of Devriendt and Lambiotte. Analogously, a graph may be resistance positive or RP; we say a graph is strictly RN if it is RN but not RP. In this paper, we show that every $2$-connected strictly RN graph is bipartite with parts whose sizes differ by one, demonstrating that there are no $1$-tough strictly RN graphs. As a consequence, we prove that every $1$-tough RN graph is also RP. Lastly, we quantify the exact toughness values an RN graph can attain below $1$.

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Cite

BibTeXRIS
BibTeX
@misc{garcia2026classification,
  title = {{Classification of strictly resistance nonnegative graphs}},
  author = {Hailey Jay Garcia},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23394},
  eprint = {2609.23394},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23394},
}
RIS
TY  - UNPB
TI  - Classification of strictly resistance nonnegative graphs
AU  - Hailey Jay Garcia
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23394
UR  - https://arxiv.org/abs/2609.23394
AB  - We say that a graph is resistance nonnegative or RN if it admits a positive edge-weight that yields nonnegative resistance curvature in the sense of Devriendt and Lambiotte. Analogously, a graph may be resistance positive or RP; we say a graph is strictly RN if it is RN but not RP. In this paper, we show that every $2$-connected strictly RN graph is bipartite with parts whose sizes differ by one, demonstrating that there are no $1$-tough strictly RN graphs. As a consequence, we prove that every $1$-tough RN graph is also RP. Lastly, we quantify the exact toughness values an RN graph can attain below $1$.
ER  -