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

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 $M(T,\varepsilon)x=\varepsilon+αe_r$ 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.