Kadison–Singer Methods for Ihara Zeta Functions

Definitions and mathematical statement

The supplied question concerns the Ihara zeta function of a connected regular graph. The record does not explicitly specify finiteness, directedness, or whether loops and multiple edges are allowed. The following definitions give a conditional formulation under the conventional finite, undirected, simple graph interpretation; these conventions require confirmation against the source.

Let $G=(V,E)$ be a finite, connected, undirected, simple $k$-regular graph, where $k\in{\mathbb{Z}}_{\geq 0}$, meaning that every vertex has degree $k$. Put $n=|V|$ and $m=|E|$. Let $A\in\{0,1\}^{n\times n}$ be its adjacency matrix, and let $I_n$ be the identity matrix. Replace each edge by its two orientations, and denote the reverse of an oriented edge $e$ by $\bar e$.

A closed oriented edge walk $C=(e_1,\ldots,e_\ell)$ is cyclically nonbacktracking if successive edges are incident in the direction of travel and $e_{i+1}\ne\bar e_i$ for every $i$, with indices read modulo $\ell$. It is primitive if it is not a repetition of a shorter closed walk. Let $\mathcal P(G)$ be the set of equivalence classes of primitive cyclically nonbacktracking closed walks under cyclic rotation; reversal is not an additional equivalence. Write $\ell(C)$ for the length of a representative. Define the Ihara zeta function as the formal power series \[ \zeta(u,G)=\prod_{[C]\in\mathcal P(G)}\left(1-u^{\ell(C)}\right)^{-1}\in{\mathbb{Q}}[[u]]. \] The determinant identity supplied in the record is \[ \zeta(u,G)^{-1}=(1-u^2)^{m-n}\det\!\left(I_n-uA+(k-1)u^2I_n\right). \] Here the formal power series interpretation avoids imposing an unspecified analytic domain for $u$.

Source research question. Can techniques from the Kadison--Singer workshop, described in the supplied expanded statement as Kadison--Singer and interlacing-polynomial methods, yield further information about $\zeta(u,G)$ for connected regular graphs?

The determinant identity is supplied background, rather than the requested new conclusion. Neither the original question nor its expanded formulation specifies a property of the zeta function to establish, a proposed bound, or the intended quantification over graphs and degrees. Thus the record defines a research direction but does not determine a single precise conjecture with a truth value. No stronger target statement is inferred here.

Short English statement

The question asks whether techniques discussed at the Kadison--Singer workshop, including interlacing-polynomial methods in the expanded record, can provide further information about the Ihara zeta function of a connected regular graph. The supplied determinant formula describes the object, but the desired new conclusion and some graph conventions remain unspecified. The supplied open label is a dated curation assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002847$, title: Kadison--Singer Methods for Ihara Zeta Functions; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. No individual proposer is named.
• Supplied report citation: Arbeitsgemeinschaft: The Kadison-Singer Conjecture (2016), Oberwolfach Reports, pp. 927--969, DOI 10.4171/owr/2015/17. Supplied source URL and literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2015/17. The background attributes the question to workshop open-problem discussions.
• The registry gives proposed year $2015$ and an owr-year:2015 tag, while the supplied report citation includes $2016$. This discrepancy is retained without resolution.
• Literature assessment dated exactly 2026-08-21: the supplied status is open. The supplied assessment says that the extract asks broadly for further information without specifying a proposition or target theorem, and that no definite resolution is identified. This is supplied dated curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The supplied statement-status label is corrected_verified. Its accompanying verification note reports that the official Oberwolfach passage was checked directly and that the question, definitions, and notation were restored without adding a stronger claim. This is a supplied provenance claim; no source visit or independent verification was performed for this draft.