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PreprintCC BY-NC-ND 4.0arXiv

Isomorphism Criterion of Monomial Digraphs over Prime Fields

  • Alexander M. Kodess
  • Felix Lazebnik
  • Mikhail Muzychuk
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Abstract

For any Galois field 𝔽q with q elements and any positive integers m and n, the directed graph D(q;m,n) has vertex set 𝔽q×𝔽q, and there is an arc from vertex (x1,x2) to vertex (y1,y2) if and only if x2+y2=x1my1n. It was conjectured in earlier work that two digraphs D(q;m1,n1) and D(q;m2,n2) are isomorphic if and only if there exists an integer k relatively prime to q−1 such that m2≡ km1 and n2 ≡ kn1, where both congruences are modulo q−1. We prove this conjecture over prime fields and construct an infinite family of counterexamples over extension fields.

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Cite

BibTeXRIS
BibTeX
@misc{kodess2026isomorphism,
  title = {{Isomorphism Criterion of Monomial Digraphs over Prime Fields}},
  author = {Alexander M. Kodess and Felix Lazebnik and Mikhail Muzychuk},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06735},
  eprint = {2610.06735},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06735},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Isomorphism Criterion of Monomial Digraphs over Prime Fields
AU  - Alexander M. Kodess
AU  - Felix Lazebnik
AU  - Mikhail Muzychuk
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06735
UR  - https://arxiv.org/abs/2610.06735
AB  - For any Galois field 𝔽_q with q elements and any positive integers m and n, the directed graph D(q;m,n) has vertex set 𝔽_q×𝔽_q, and there is an arc from vertex (x_1,x_2) to vertex (y_1,y_2) if and only if x_2+y_2=x_1^my_1^n. It was conjectured in earlier work that two digraphs D(q;m_1,n_1) and D(q;m_2,n_2) are isomorphic if and only if there exists an integer k relatively prime to q−1 such that m_2≡ km_1 and n_2 ≡ kn_1, where both congruences are modulo q−1. We prove this conjecture over prime fields and construct an infinite family of counterexamples over extension fields.
ER  -