Diagonal parity and loop toggling for symmetric matrices over 𝔽2
Abstract
Let M be a symmetric matrix over 𝔽2, and let diag(M) be its diagonal vector. It is known that diag(M)∈Im(M). Thus the affine system Mx=diag(M) is always solvable. We strengthen this existence statement to a parity rigidity theorem: every solution satisfies diag(M)T x≡rank(M)pmod2. For graph matrices, this gives a common extension of Sutner's odd-domination theorem and Batal's parity theorem from closed-neighborhood matrices A(G)+I to arbitrary partially looped graph matrices A(G)+D. We also study how rank and nullity change when loops are toggled. Algebraically, simultaneous loop toggling on the support of a vector u is the diagonal rank-one update M↦ M+uuT. We prove an exact three-case rank and nullity formula for this update. Finally, for rooted trees with arbitrary binary diagonal labels, we give a finite-state boundary recursion using affine subspaces of 𝔽22. This recursion counts all generalized odd-domination patterns and implies eventual quasigeometric nullity formulas for complete rooted trees with eventually periodic depth labels.
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BibTeX
@misc{aliabadi2026diagonal,
title = {{Diagonal parity and loop toggling for symmetric matrices over $\mathbb F_2$}},
author = {Mohsen Aliabadi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.11056},
eprint = {2605.11056},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.11056},
}
RIS
TY - UNPB
TI - Diagonal parity and loop toggling for symmetric matrices over 𝔽_2
AU - Mohsen Aliabadi
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2605.11056
UR - https://arxiv.org/abs/2605.11056
AB - Let M be a symmetric matrix over 𝔽_2, and let diag(M) be its diagonal vector. It is known that diag(M)∈Im(M). Thus the affine system Mx=diag(M) is always solvable. We strengthen this existence statement to a parity rigidity theorem: every solution satisfies diag(M)^T x≡rank(M)pmod2. For graph matrices, this gives a common extension of Sutner's odd-domination theorem and Batal's parity theorem from closed-neighborhood matrices A(G)+I to arbitrary partially looped graph matrices A(G)+D. We also study how rank and nullity change when loops are toggled. Algebraically, simultaneous loop toggling on the support of a vector u is the diagonal rank-one update M↦ M+uu^T. We prove an exact three-case rank and nullity formula for this update. Finally, for rooted trees with arbitrary binary diagonal labels, we give a finite-state boundary recursion using affine subspaces of 𝔽_2^2. This recursion counts all generalized odd-domination patterns and implies eventual quasigeometric nullity formulas for complete rooted trees with eventually periodic depth labels.
ER -