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PreprintarXiv licencearXiv

Resolution of a problem of Mohar on non-positive inertia

  • Clive Elphick
  • Hitesh Kumar
  • Shivaramakrishna Pragada
  • Thomás Jung Spier
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Abstract

For a graph G of order n, its positive, negative and non-positive inertia is the number of positive, negative and non-positive eigenvalues of its adjacency matrix A(G), respectively. Mohar asked whether every graph with k non-positive eigenvalues has order O(k2) as k→ ∞. Using NEPS, we construct a sequence of non-singular graphs with negative inertia k and order Ω(k7/3) as k→ ∞, thus resolving Mohar's problem. Our result also strongly refutes a recent conjecture of Akbari, Elphick, Kumar, Pragada, and Tang involving positive and negative inertia.

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Cite

BibTeXRIS
BibTeX
@misc{elphick2026resolution,
  title = {{Resolution of a problem of Mohar on non-positive inertia}},
  author = {Clive Elphick and Hitesh Kumar and Shivaramakrishna Pragada and Thomás Jung Spier},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06319},
  eprint = {2609.06319},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06319},
}
RIS
TY  - UNPB
TI  - Resolution of a problem of Mohar on non-positive inertia
AU  - Clive Elphick
AU  - Hitesh Kumar
AU  - Shivaramakrishna Pragada
AU  - Thomás Jung Spier
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06319
UR  - https://arxiv.org/abs/2609.06319
AB  - For a graph G of order n, its positive, negative and non-positive inertia is the number of positive, negative and non-positive eigenvalues of its adjacency matrix A(G), respectively. Mohar asked whether every graph with k non-positive eigenvalues has order O(k^2) as k→ ∞. Using NEPS, we construct a sequence of non-singular graphs with negative inertia k and order Ω(k^7/3) as k→ ∞, thus resolving Mohar's problem. Our result also strongly refutes a recent conjecture of Akbari, Elphick, Kumar, Pragada, and Tang involving positive and negative inertia.
ER  -