Matrix Liberation Lemma

• Matrix Liberation Lemma is a criterion in spectral graph theory that allows the simultaneous alteration of zero entries in symmetric matrices while maintaining eigenvalue multiplicities and the strong spectral property.
• It provides both algebraic (vector-based) and combinatorial (set-based) methods to verify conditions and facilitate perturbations, advancing solutions to the inverse eigenvalue problem for graphs.
• Practical applications include direct sum constructions, zero forcing in Cartesian products, and resolving spectral arbitrariness in six-vertex graphs.

The Matrix Liberation Lemma is a criterion in spectral graph theory for simultaneously altering multiple zero entries of a symmetric matrix (with prescribed off-diagonal sparsity pattern) while maintaining prescribed spectral or rank properties. It provides a unified perspective that encompasses previous incremental edge-perturbation lemmata and generalizes the direct sum constructions, particularly when eigenvalue multiplicities coincide. Recent advances have yielded both an algebraic (vector-based) and a combinatorial (set-based) version of the lemma, enabling efficient verification and application in resolving previously open cases of the inverse eigenvalue problem of graphs.

1. Preliminaries and Vector-Version of the Matrix Liberation Lemma

For a simple graph $G$ with $n$ vertices, the set $S(G)$ consists of all real symmetric matrices where $A_{ij} \neq 0$ if and only if $\{i, j\} \in E(G)$ for $i \neq j$. The strong spectral property (SSP) for $A \in S(G)$ is defined such that the only symmetric matrix $X$ satisfying $A \circ X = I \circ X = 0$ and $[A, X] = 0$ is $X = 0$, where $\circ$ denotes entrywise product and $[A, X]$ is the commutator. The verification (Jacobian) matrix $\Psi(A)$ is constructed by vectorizing commutators $[A, X^{ij}]$ across pairs $\{i, j\} \in E(\overline{G})$. The kernel of $\Psi(A)^\top$ characterizes the perturbations preserving the spectrum and sparsity constraints.

Barrett et al. (2020) formalized the Matrix Liberation Lemma (vector version):

• Let $A \in S(G)$.
• If a vector $x \in \mathrm{Col}(\Psi(A))$ with support $\beta \subseteq E(\overline{G})$ exists such that $A$ retains SSP with respect to $G + \beta$, then a perturbed matrix $A' \in S(G+\beta)$ with $\mathrm{spec}(A') = \mathrm{spec}(A)$ and SSP w.r.t. $G+\beta$ is constructible.
• The process involves constructing $\Psi(A)$, selecting an appropriate support vector, and applying a perturbative argument.

2. Combinatorial Set-Version and Liberation Sets

Lin, Oblak, and Šmigoc reframed the lemma using a purely combinatorial notion: for $A \in S(G)$, a nonempty $\beta \subseteq E(\overline{G})$ is an SSP liberation set if, for every $\beta' \subset \beta$ with $|\beta'| = |\beta| - 1$, $A$ has the SSP for $G+\beta'$. The set-version of the Matrix Liberation Lemma states that if $\beta$ is a liberation set, there exists $A' \in S(G+\beta)$ with $\mathrm{spec}(A') = \mathrm{spec}(A)$ and SSP on $G+\beta$.

This equivalence passes through a linear algebra result: $\beta$ is a liberation set if and only if a vector in $\mathrm{Col}(\Psi(A))$ exists with support $\beta$, and required SSP conditions are met. The set-version reduces application complexity by requiring only combinatorial checking of certain submatrices for full row-rank, rather than explicit witness construction.

3. Proof Outline and Technical Implementation

The set-version proof follows three main steps:

1. Demonstrate equivalence between the liberation set condition and the full row-rank of $\Psi(A)$ submatrices formed by $E(\overline{G}) \setminus \beta \cup \{e\}$ for each $e \in \beta$.
2. Apply a linear algebra lemma ensuring these full-rank conditions imply existence of a vector in $\mathrm{Col}(\Psi(A))$ with support precisely $\beta$.
3. Utilize the perturbation argument of Barrett–Hall–Hoover–Young to construct $A' \in S(G+\beta)$ preserving the required spectral/rank properties.

A notable implication is that the set-version enables simultaneous activation of any liberation set of edges, simplifying the verification for larger graphs and direct sum constructions.

4. Applications: Direct Sums, Zero Forcing, and Graph Constructions

4.1 Direct Sums

Given $A \in S(G)$ and $B \in S(H)$ (both satisfying SSP), consider $A \oplus B$ where the spectra intersect at $\lambda$ with multiplicities $\mult_A(\lambda)=k$, $\mult_B(\lambda)=\ell$. If the corresponding eigenspaces are generic, a rectangular grid $\beta$ formed from $V_G \subset V(G)$, $|V_G|=k$, and $V_H \subset V(H)$, $|V_H|=\ell+1$ (or $k+1$, $\ell$), is an SSP liberation set. Thus, there exists $C \in S(G \cup H + \beta)$ with $$\mathrm{spec}(C) = \mathrm{spec}(A) \cup \mathrm{spec}(B)$$ and the SSP.

Example: For two trees on six vertices $K_{1,3}$ and $P_2$ with spectra $\{0, \theta^{(2)},1\}$ and $\{\theta,2\}$, joining three leaves to both vertices suffices to construct an SSP-matrix on the augmented graph with spectrum $\{0, \theta^{(3)},1,2\}$, rendering multiplicity list $(1,2,2,1)$ spectrally arbitrary on $G_{100}$.

4.2 Zero Forcing and Cartesian Products

For $G \square H$ (Cartesian product), if $\beta'$ is a zero-forcing set, then for all SSP-matrices $A \in S(G)$, $B \in S(H)$, the direct sum $A \oplus B$ has SSP on $(G \square H) + \beta'$. If $\beta$ is a zero-forcing cover—removal of any vertex leaves a zero-forcing set—then $\beta$ is a liberation set by Theorem 5.6.

Example: For $G = P_s$, $H = P_t$, $Z(P_s \square P_t) = \min(s, t)$. A zero-forcing cover of size $\min(s, t) + 1$ yields spectrally arbitrary matrices for ladder and mesh graphs. For the prism $G = C_4$, $H = P_2$, a 4-vertex zero-forcing cover yields an SAP-liberation set ensuring 3-fold nullity on $C_4 P_2 + \beta$.

5. Comparative Analysis with Earlier Techniques

Earlier lemmata, such as the Supergraph Lemma and the Direct-Sum Lemma, could guarantee spectrum preservation only under restrictive conditions (single edge perturbation at a time, or direct sum with disjoint spectra, respectively). The Matrix Liberation Lemma generalizes these approaches by simultaneously activating bundles of zero entries and handling overlapping eigenvalue multiplicities if eigenspaces are generic. This suggests substantially increased flexibility in solving the inverse eigenvalue problem for complex graphs.

6. Resolution of Six-Vertex Graph Spectral Arbitraryness

Application of the Matrix Liberation Lemma in both the direct sum/generic eigenspace context and zero-forcing cover arguments has resolved outstanding open multiplicity-arbitrariness cases for all six-vertex graphs, as detailed in Lin–Oblak–Šmigoc (Lin et al., 2023). Graphs such as $G_{100}$, $G_{127}$, $G_{169}$, $G_{129}$, $G_{145}$, $G_{153}$, $G_{151}$, $G_{163}$, $G_{171}$, $G_{175}$, and $G_{187}$ are now proven spectrally arbitrary for their respective ordered multiplicity lists, completing precedently unresolved groups in Barioli–Fallat–Hogben–Young (2018).

Graph	Ordered Multiplicity Lists Resolved	Example Reference
$\mathsf{G_{100}}$	$(1,2,2,1)$	Example 4.5
$\mathsf{G_{127}}, \mathsf{G_{169}}$	$(2,1,1,2), (3,2,1)$	Example 4.10
$\mathsf{G_{129}},\mathsf{G_{145}},\mathsf{G_{153}}$	$(1,1,3,1), (1,3,1,1)$	Example 5.14
$\mathsf{G_{151}}$	$$(1,1,3,1), (1,3,1,1), (1,2,3), (3,2,1), (1,3,2), (2,3,1)$$	Example 4.7
$\mathsf{G_{163}}$	$(1,1,3,1), (1,3,1,1)$	Example 4.11

A plausible implication is that the combinatorial perspective afforded by the set-version of the Matrix Liberation Lemma will continue to streamline resolution of spectral arbitrariness in higher-order graph families and further generalizations of the inverse eigenvalue problem.