Diagonal parity and loop toggling for symmetric matrices over $\mathbb F_2$

TL;DR

This paper extends parity properties of adjacency matrices over GF(2) to more general symmetric matrices with diagonal perturbations, providing new rank formulas, toggling loop effects, and recursive solution counting for trees.

Contribution

It generalizes known parity phenomena to arbitrary symmetric matrices with diagonal modifications, including new rank formulas and solution enumeration methods for trees.

Findings

Every diagonal perturbation solution satisfies a rank parity relation.

Rank-one diagonal toggling affects solution spaces predictably.

Recursive formulas count solutions in rooted trees with periodic diagonal labels.

Abstract

Let $G$ be a finite simple graph with adjacency matrix $A(G)$ over $\mathbb F_2$. The closed neighborhood matrix $A(G)+I$ is central in the theory of odd domination. Sutner proved that every graph has an odd dominating set, equivalently $\mathbf 1$ lies in the range of $A(G)+I$, and Batal proved that every such set has cardinality congruent to $\rank(A(G)+I)$ modulo $2$. We extend this parity phenomenon from closed neighborhood matrices to partially looped graph matrices $A(G)+D$, where $D$ is an arbitrary diagonal matrix over $\mathbb F_2$. Equivalently, we work with arbitrary symmetric matrices $M$ over $\mathbb F_2$ and the natural right-hand side $\diag(M)$. We include a self-contained proof, attributed by Filmus to Alon, that $\diag(M)\in\Img(M)$, and we prove that every solution of $Mx=\diag(M)$ satisfies \[ \diag(M)^\top x\equiv \rank(M)\pmod 2. \] We also give a complete rank and nullity formula for rank-one diagonal perturbations $M\mapsto M+uu^\top$, which in the graph setting describes exactly how toggling loops changes the associated solution spaces. Finally, for rooted trees with arbitrary diagonal labels, we develop a finite-state boundary recursion that counts all solutions of $M(T,\varepsilon)x=\varepsilon+\alpha e_r$ with prescribed root value, and we derive explicit nullity formulas for complete rooted $d$-ary trees. For $d\ge2$, we also prove an eventual-periodicity theorem for complete rooted $d$-ary trees with depth-dependent eventually periodic diagonal labels.