Odd-Symmetric Formulation

• Odd-symmetric formulation is a linear-algebraic framework for symmetric matrices over F2 that couples a matrix with its diagonal to enforce a rigid parity invariant linked to the matrix rank.
• It unifies concepts from graph theory and linear algebra by equating closed-neighborhood odd domination with the solvability of Mx = diag(M) in both partially looped graphs and arbitrary symmetric matrices.
• The formulation offers explicit methods for handling diagonal perturbations and rooted tree recursions, yielding precise rank, nullity, and solution-space characterizations.

Odd-symmetric formulation denotes a linear-algebraic framework for symmetric matrices over $\mathbb F_2$ in which a matrix $M$ is paired with its diagonal vector $d=\operatorname{diag}(M)$ and studied through the system $Mx=d$. In graph-theoretic language this encompasses closed-neighborhood odd domination, partially looped graph matrices $A(G)+\operatorname{diag}(\varepsilon)$, and arbitrary symmetric matrices over $\mathbb F_2$. Its defining feature is a rigid parity law: the solvable system $Mx=\operatorname{diag}(M)$ has solution sets whose global parity invariant is fixed by $\operatorname{rank}(M)$, while loop toggling and rooted-tree recursions admit explicit structural descriptions (Aliabadi, 11 May 2026).

1. From odd domination to symmetric matrices

The starting point is a finite simple graph $G$ with adjacency matrix $A(G)$ over $M$0. The closed neighborhood matrix is

$M$1

For a subset $M$2, with characteristic vector $M$3, the equation

$M$4

is equivalent to the condition that every closed neighborhood $M$5 meets $M$6 in odd cardinality. Such an $M$7 is an odd dominating set. In this classical setting, Sutner proved that $M$8, and Batal proved that every solution $M$9 of $d=\operatorname{diag}(M)$0 satisfies

$d=\operatorname{diag}(M)$1

(Aliabadi, 11 May 2026).

The odd-symmetric formulation extends this picture in two directions. First, for a labeling $d=\operatorname{diag}(M)$2, one defines

$d=\operatorname{diag}(M)$3

which may be viewed as adding a loop at $d=\operatorname{diag}(M)$4 when $d=\operatorname{diag}(M)$5. The associated parity equation becomes

$d=\operatorname{diag}(M)$6

Second, every symmetric matrix $d=\operatorname{diag}(M)$7 can be regarded as the adjacency matrix of a possibly looped graph, and the natural right-hand side is its diagonal

$d=\operatorname{diag}(M)$8

The central system is then

$d=\operatorname{diag}(M)$9

In this formulation, the diagonal is not an arbitrary inhomogeneity. It is intrinsic to the matrix and encodes the loop pattern of the underlying graph. The terminology “odd-symmetric” refers to the way diagonal data and symmetry over $Mx=d$0 combine to produce a uniform parity phenomenon across all solutions.

2. Fundamental parity law

The first structural statement is solvability. If $Mx=d$1 is symmetric over $Mx=d$2 and $Mx=d$3, then

$Mx=d$4

A self-contained proof proceeds from the identity $Mx=d$5 for symmetric $Mx=d$6. For any $Mx=d$7,

$Mx=d$8

Over $Mx=d$9, the off-diagonal contributions in $A(G)+\operatorname{diag}(\varepsilon)$0 cancel in pairs, so

$A(G)+\operatorname{diag}(\varepsilon)$1

Hence $A(G)+\operatorname{diag}(\varepsilon)$2 for all $A(G)+\operatorname{diag}(\varepsilon)$3, which implies $A(G)+\operatorname{diag}(\varepsilon)$4. The affine solution set

$A(G)+\operatorname{diag}(\varepsilon)$5

is therefore always nonempty (Aliabadi, 11 May 2026).

The central theorem states that $A(G)+\operatorname{diag}(\varepsilon)$6 is a nonempty affine subspace of dimension $A(G)+\operatorname{diag}(\varepsilon)$7, and every solution $A(G)+\operatorname{diag}(\varepsilon)$8 satisfies

$A(G)+\operatorname{diag}(\varepsilon)$9

The identity $\mathbb F_2$0 again uses cancellation of off-diagonal terms. The rigidity of $\mathbb F_2$1 across the whole fiber $\mathbb F_2$2 follows from the fact that, for symmetric $\mathbb F_2$3, if $\mathbb F_2$4 and $\mathbb F_2$5, then $\mathbb F_2$6.

To identify this constant with $\mathbb F_2$7, the argument invokes the classification of symmetric bilinear forms over $\mathbb F_2$8. Writing $\mathbb F_2$9, one decomposes

$Mx=\operatorname{diag}(M)$0

where $Mx=\operatorname{diag}(M)$1, each $Mx=\operatorname{diag}(M)$2 is one-dimensional with Gram matrix $Mx=\operatorname{diag}(M)$3, and each $Mx=\operatorname{diag}(M)$4 is a hyperbolic plane with Gram matrix

$Mx=\operatorname{diag}(M)$5

Then

$Mx=\operatorname{diag}(M)$6

and one constructs a vector $Mx=\operatorname{diag}(M)$7 such that $Mx=\operatorname{diag}(M)$8 for all $Mx=\operatorname{diag}(M)$9. Any solution $\operatorname{rank}(M)$0 represents the same functional as $\operatorname{rank}(M)$1, so $\operatorname{rank}(M)$2, and

$\operatorname{rank}(M)$3

This parity law is the core of the formulation. The scalar $\operatorname{rank}(M)$4, described in the paper as the mod-$\operatorname{rank}(M)$5 “energy” $\operatorname{rank}(M)$6, is independent of the chosen solution and depends only on $\operatorname{rank}(M)$7.

3. Graph-theoretic interpretation

For a partially looped graph matrix

$\operatorname{rank}(M)$8

the diagonal is exactly $\operatorname{rank}(M)$9, and the system

$G$0

can be read vertexwise as

$G$1

Thus each vertex contributes one of two parity constraints. If $G$2, the equation is

$G$3

so the closed neighborhood of $G$4 is odd. If $G$5, the equation is

$G$6

so the open neighborhood of $G$7 is even.

A solution $G$8 may therefore be viewed as an $G$9-odd dominating pattern: a vertex labeling or subset satisfying a mixture of closed-neighborhood oddness and open-neighborhood evenness, determined locally by the loop pattern $A(G)$0. The global law becomes

$A(G)$1

Combinatorially, $A(G)$2 is the parity of the number of vertices that are both looped and selected. No matter which solution pattern is chosen, that parity is forced by the rank.

The classical Sutner–Batal theorem is recovered by taking $A(G)$3. Then

$A(G)$4

and the general theorem specializes to

$A(G)$5

Existence of odd dominating sets is exactly the statement $A(G)$6.

This suggests that odd domination is only one instance of a broader structural law: the local choice of open versus closed parity conditions can vary from vertex to vertex without losing either solvability or the rank-controlled parity invariant.

4. Loop toggling and rank-one perturbations

The formulation includes an exact calculus for changing the diagonal. For any symmetric $A(G)$7 and $A(G)$8, consider the rank-one diagonal perturbation

$A(G)$9

In graph language, the special case $M$00 toggles the loop at a single vertex $M$01.

If $M$02, then

$M$03

so the kernel drops by codimension one: $M$04

If $M$05, choose $M$06 with $M$07. The scalar $M$08 is independent of the choice of $M$09. There are then two subcases. If $M$10, the perturbation is invisible to rank and nullity: $M$11 If $M$12, then

$M$13

with $M$14, and

$M$15

For loop toggling at a vertex $M$16, set $M$17. Then the rank changes by $M$18, $M$19, or $M$20, according to whether $M$21, or $M$22 with the distinguished scalar $M$23 equal to $M$24 or $M$25. The effect on the solution space of

$M$26

is therefore completely controlled by elementary image–kernel data (Aliabadi, 11 May 2026).

A plausible implication is that the odd-symmetric framework is not merely existential. It also provides a stepwise mechanism for traversing the family of matrices $M$27 by changing one loop at a time while tracking rank, nullity, and solution-space dimension exactly.

5. Rooted trees and finite-state boundary recursion

For rooted trees, the formulation becomes explicitly recursive. Let $M$28 be a rooted tree with root $M$29 and labeling $M$30, and write

$M$31

For $M$32, one studies

$M$33

Here $M$34 introduces an inhomogeneity at the root equation, and $M$35 prescribes the root variable. The counting function

$M$36

is assembled into the boundary enumerator

$M$37

Every rooted labeled tree falls into exactly one of three boundary types. There exist a type $M$38 and an integer $M$39 such that: $M$40 or

$M$41

or

$M$42

The behavior at the root is thus encoded by a finite-state system with only three states.

If the root has child subtrees $M$43 with weighted types $M$44, and

$M$45

then the parent type is determined recursively. If some child has type $M$46, then

$M$47

If no child has type $M$48, then

$M$49

From these three types one reads off the nullity of the odd-symmetric system $M$50: $M$51

The recursion yields explicit formulas for complete rooted $M$52-ary trees $M$53. For uniform labels $M$54,

$M$55

For uniform labels $M$56 and even $M$57,

$M$58

For uniform labels $M$59 and odd $M$60, the nullity follows a 3-phase pattern: $M$61

$M$62

$M$63

The paper also proves an eventual-periodicity theorem for complete rooted $M$64-ary trees with depth-dependent eventually periodic diagonal labels. For $M$65, if the label sequence $M$66 is eventually periodic, then the type sequence $M$67 is eventually periodic, and on each residue class modulo a period $M$68,

$M$69

for all sufficiently large $M$70 (Aliabadi, 11 May 2026).

6. Scope and significance

The odd-symmetric formulation unifies four statements that are separate in the classical odd-domination literature. First, for every symmetric matrix $M$71 over $M$72, the system

$M$73

is solvable. Second, every solution has the same mod-$M$74 signature,

$M$75

Third, diagonal changes of the form $M$76 admit complete rank and nullity formulas. Fourth, on rooted trees the solution spaces can be counted by a finite-state boundary recursion and analyzed asymptotically (Aliabadi, 11 May 2026).

In graph-theoretic terms, this shifts the emphasis from the special closed-neighborhood matrix $M$77 to the entire family $M$78 of partially looped graph matrices. The diagonal encodes where oddness is imposed locally, symmetry enforces cancellation in quadratic forms, and rank supplies a global invariant that every solution must satisfy. What begins as an existence-and-parity theorem for odd dominating sets thereby becomes a structural law for arbitrary symmetric matrices over $M$79.

This suggests a broader conceptual interpretation. The formulation is “odd-symmetric” not because it introduces a new symmetry class, but because it isolates a characteristic-$M$80 interaction between diagonal data, symmetric bilinear structure, and parity. In that setting, solvability is automatic, solution parity is rigid, diagonal perturbations are tractable, and recursive combinatorics emerge naturally on trees.