Combinatorial Formulas for Graph Zeta Functions

Definitions and mathematical statement

Let $V$ be the fixed vertex set in the source and let $\Gamma=(V,E)$ range over the intended class of graphs on $V$. Write $W^-_\Gamma(X,T)$ for the graph-associated function named in the source, with formal variables $X,T$. The supplied record does not define this function or specify its ambient function ring. It also does not explicitly specify the graph conventions or permitted vertex-set sizes; the appearance of $[n]=\{1,\ldots,n\}$ suggests finite vertex sets, but the treatment of the empty vertex set is unstated.

Under the usual finite simple undirected graph convention, a cograph is a graph obtained from single-vertex graphs by repeated disjoint union and join, where the join adds every edge between the two constituent vertex sets. This convention explains the standard term but does not resolve the missing graph conventions in the record.

Requested statement. For each fixed vertex set $V$, find a combinatorial expression for $W^-_\Gamma(X,T)$, uniform over all graphs $\Gamma$ on $V$, which extends the modelling-hypergraph formula supplied for cographs. The source asks that the expression be "meaningful" and in the spirit of its Theorem 1; it supplies no formal criterion for meaningfulness or for what constitutes an extension.

The supplied extract attributes Theorem 1 to Rossmann--Voll, reference [5], Theorems C & D. It reports that each cograph $\Gamma=(V,E)$ has a modelling hypergraph written $H=(V,(\mu_I)_{I\subset V})$, and that, for every compact discrete valuation ring $\mathcal O$ with residue-field cardinality $q$, a graphical zeta function is obtained by evaluating the corresponding hypergraph expression at $(X,T)=(q,q^{-s})$. The meanings and allowed values of $\mu_I$, the precise subset-indexing convention, the graphical zeta function, and its domain in $s$ are not defined in the record. The extracted phrase "compact ask DVR" is corrupted or unexplained. The displayed formula (1) is also corrupted, so its exponents, factors, and summation conventions cannot be transcribed reliably.

The source proposes relating $\Delta(2^{[n]})$ to $L(P_n)$, described respectively as an order complex and the face lattice of a permutohedron. For a poset, its order complex has finite chains as faces. However, the subset-poset conventions, empty-face conventions, and permutohedron indexing needed for the intended formula are not supplied. The background further mentions sandwich comparisons and a reformulation for families of graphs, without giving their definitions or the family-level statement. These qualifications remain unresolved; this is therefore a statement-only draft of the research target, not a self-contained definition of the zeta functions or a reconstruction of formula (1).

Short English statement

Find a uniform combinatorial formula for the supplied graph zeta functions $W^-_\Gamma(X,T)$, for all graphs on a fixed vertex set, extending the reported cograph formula. The function definitions, graph conventions, and a reliable transcription of that formula are missing. The supplied open status is dated curation and requires independent verification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006430$, title Combinatorial Formulas for Graph Zeta Functions, exact source problem code catalogue entry. The record supplies no named proposer and records proposed year $2025$.
• Supplied report citation: Cohomology Theories for Automorphic Forms and Enumerative Algebra (2026), Oberwolfach Reports, pp. 2065--2124, DOI 10.4171/owr/2025/38. https://doi.org/10.4171/owr/2025/38. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Oberwolfach Report 38. https://ems.press/content/serial-article-files/52251.
• Supplied literature source: Earlier graphical zeta results. https://ems.press/content/serial-article-files/46914. The extract attributes its Theorem 1 to Rossmann--Voll, reference [5], Theorems C & D; the bibliographic entry for [5] is not supplied, and its correspondence with this link is not established by the record.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, reports a combinatorial Denef formula for cographs, describes sandwich comparisons as an obstruction in the 2025 discussion, mentions a reformulation for graph families, and states that no formula valid for all graphs is known. This is supplied dated curation, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that only formula delimiters, notation, grammar, and typography were repaired while preserving mathematical content and scope. This supplied editorial label does not resolve the missing definitions or the corrupted displayed formula.