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PreprintarXiv licencearXiv

So's Conjecture for Integral Circulant Graphs of Order p^aq

  • Jianwei Jiang
  • Chunhua Yang
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Abstract

So conjectured that, for a fixed positive integer n, the ordinary adjacency spectrum of an integral circulant graph of order n determines its divisor set. We prove this for graphs of order p to the power a times q, where p and q are primes with p less than q and a is at least one. To handle coincident eigenvalues arising from distinct greatest-common-divisor classes, we use a spectral counting measure. For connected graphs, an exact identity recovers the part of the divisor set consisting of one and q, when present, together with the counting measure for a graph of order p to the power a minus one times q. Strong induction and decomposition into connected components then recover the full divisor set, including the case p equals two and disconnected graphs.

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Cite

BibTeXRIS
BibTeX
@misc{jiang2026conjecture,
  title = {{So's Conjecture for Integral Circulant Graphs of Order p^aq}},
  author = {Jianwei Jiang and Chunhua Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20072},
  eprint = {2609.20072},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20072},
}
RIS
TY  - UNPB
TI  - So's Conjecture for Integral Circulant Graphs of Order p^aq
AU  - Jianwei Jiang
AU  - Chunhua Yang
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20072
UR  - https://arxiv.org/abs/2609.20072
AB  - So conjectured that, for a fixed positive integer n, the ordinary adjacency spectrum of an integral circulant graph of order n determines its divisor set. We prove this for graphs of order p to the power a times q, where p and q are primes with p less than q and a is at least one. To handle coincident eigenvalues arising from distinct greatest-common-divisor classes, we use a spectral counting measure. For connected graphs, an exact identity recovers the part of the divisor set consisting of one and q, when present, together with the counting measure for a graph of order p to the power a minus one times q. Strong induction and decomposition into connected components then recover the full divisor set, including the case p equals two and disconnected graphs.
ER  -