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4895 bytes, as of the version from 2026-10-29 12:06 (current). Recorded here, not run.
# Argument selection for T608
The sole free parameter is `g`, a graph isomorphism class, of category
`discrete_index`. The domain consists of finite, simple, undirected, connected
graphs whose minimum vertex degree is at least two. The quantity is the ordinary
unweighted reciprocal Ihara zeta polynomial in a formal variable. There is no
chosen evaluation point, continuous coupling, additional quantity, or hidden
probability parameter.
The graph is an intrinsic input of this established graph invariant. Its prime
walk interpretation, edge-matrix determinant and Bass determinant are documented
in I. Sato, *The Ihara Zeta function and Quantum Walk* (2019), pp. 202–204,
and Hasegawa–Saito (2015), pp. 125–128. Orientation is retained: reversing a
primitive closed walk gives a different class. Tree attachments create no
reduced primitive closed walks, which motivates excluding vertices of degree
one. This restriction does not exclude bridges within the remaining graph.
The default selection is **every eligible isomorphism class on three through
six vertices**, not a list selected for named examples or numerical coincidences.
The upper bound six is a noncanonical computational and presentation choice,
independent of numeral notation. At this size one already sees cycles, complete
and complete bipartite graphs, regular and irregular graphs, a triangular prism,
an octahedral graph, the bowtie, and two triangles joined by a bridge. The last
example first occurs on six vertices and prevents conflating minimum degree two
with being bridgeless. Complete coverage at each included order avoids an
unexplained preference for any one topology. Increasing the vertex bound adds
rapidly growing numbers of isomorphism classes, so this initial reference stops
at the first order containing these contrasting structures rather than extending
a single named family more deeply. No claim is made that six is a canonical
mathematical boundary or that every individual graph has a separate application.
Two independent enumerations agree: nauty `geng -c -d2` and the NetworkX copy
of Read and Wilson's *Atlas of Graphs*. They give 1, 3, 11 and 61 classes at
orders 3, 4, 5 and 6, respectively, totaling 76. The selection contains 10
regular graphs, 66 irregular graphs, seven bipartite graphs, and one graph
with a bridge. It includes every qualifying graph, regardless of whether a
polynomial coincides with another. In this extent all 76 polynomials differ.
This is a discrete family, not an exploratory grid relabelled as an index.
Identifiers are Sage's `canonical_label(algorithm='sage').graph6_string()`.
The fixed algorithm is part of the convention: graph6 alone encodes a labelled
graph and does not choose a canonical representative. Every key agrees with
the corresponding key of both T237 and T238. The generator computes its own
selection; `neighbor-keys.json` is only an independent compatibility check,
not an input to enumeration. Rows are ordered by vertex count, edge count,
then graph6 bytes. The infinite parameter domain contains no six-vertex cutoff;
the finite extent and its rationale belong in the completeness note.
The standard size check measures a longest written polynomial of 334 characters
and about 13.1 KiB for the values and parameter records before entry comments.
The complete dry run includes the 15 familiar-name comments. Thus the chosen
extent gives broad small-graph coverage with readable exact integer polynomials;
it is not a target entry count or a response to a storage limit.
Sources checked for selection and conventions:
- Sato, pp. 202–204, definition and determinant Theorem 3:
https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2120-10.pdf
- Hasegawa and Saito, pp. 125–128, prime walks and the Ihara–Bass formula:
https://msp.org/pjm/2015/273-1/pjm-v273-n1-p06-p.pdf
- McKay, graph6 format definition:
https://users.cecs.anu.edu.au/~bdm/data/formats.txt
- NetworkX, `graph_atlas_g`, all graphs through seven vertices:
https://networkx.org/documentation/stable/reference/generated/networkx.generators.atlas.graph_atlas_g.html
- Sage graph canonical labelling documentation and the live T238 convention:
https://raw.githubusercontent.com/sagemath/sage/develop/src/sage/graphs/generic_graph.py
and https://numberdb.org/Laplacian_polynomials_of_connected_graphs
This build used the coordinated skill archive at
`agents/runs/20261029T113821Z-build-skills/manifest.json`.
Both archived versions and the source-snapshot canonical file have SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
`skill-evidence.json` records each archived version separately.
This justification is a build artifact, not an independent acceptance. A separate
critique must check the live selection, numerical claims and generator source,
then write its own argument-review sidecar and fingerprint.