What is the meaning of the multiplicity of zero as a root of a hypergraph's (or graph's) characteristic polynomial

Definitions and mathematical statement

For a finite simple undirected graph $G$ on $n$ vertices, let $A_G$ be its adjacency matrix and \[ P_G(t)=\det(tI-A_G),\qquad m_0(G)=\operatorname{ord}_{t=0}P_G(t). \] Since $A_G$ is real symmetric, $m_0(G)=\dim\ker A_G$, its nullity. This fixes the standard graph meaning; identifying the multiplicity with its spectral definition alone does not provide the combinatorial interpretation requested by the source.

Nikiforov asks for the meaning of the multiplicity of zero as a root of a hypergraph's, or graph's, characteristic polynomial. The natural target is an informative structural or combinatorial interpretation of this multiplicity. For trees the supplied summary cites the known relation \[ m_0(T)=|V(T)|-2\nu(T), \] where $\nu(T)$ is the maximum matching size. It mentions pendant/duplicate-vertex structure for general graphs but gives no universal formula or exact classification theorem. The tree formula is not asserted for arbitrary graphs.

For a hypergraph $\mathcal H$, the formal quantity would be $\operatorname{ord}_{t=0}P_{\mathcal H}(t)$ after fixing its characteristic polynomial. The source does not define that polynomial. Tensor adjacency, matrix constructions, uniformity and weighting conventions, and tensor eigenvalue notions can produce different targets, and algebraic multiplicity need not equal the dimension of a linear kernel. No one of these conventions is silently selected. The summary refers to active hypergraph spectral-nullity literature and an undefined "apparent correlation," without precise theorems or citations.

The supplied summary describes graph aspects as largely addressed and hypergraph aspects as active/open, but its discussion supplies only the specific tree matching relation and informal general comments. It labels the item LITERATURE-SURVEY in its narrative while the structured record remains open and OPEN-TRIAGE. Thus no complete graph or hypergraph resolution of the open-ended interpretation request is asserted here.

Short English statement

Nikiforov asks for a structural meaning of zero multiplicity in characteristic polynomials. For ordinary undirected graphs this is adjacency nullity, with a matching formula for trees. The hypergraph characteristic polynomial is unspecified, and the supplied general literature comments do not provide a complete interpretation.

Sources

• Local UnsolvedMath registry, permanent ID 3100027, exact source code catalogue entry, characteristic-polynomial zero-multiplicity problem; proposer Nikiforov; status open, classification OPEN-TRIAGE. The supplied narrative additionally describes a literature-survey status.
• Cooper, Combinatorial Problems I Like, October 2020 snapshot, section "Graphs, Hypergraphs, Set Systems, Designs," source-order bullet 27, https://people.math.sc.edu/cooper/combprob.html, HTML extraction accessed July 29, 2026. The provenance says the author-maintained snapshot labels the item open and later status was not checked item by item.
• The supplied summary mentions the tree matching-nullity theorem, general pendant/duplicate structures, and work by Nikiforov and others on hypergraph spectral nullity, but gives no precise bibliographic entries or general characterization theorem.