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3818 bytes, as of the version from 2026-10-27 04:15 (current). Recorded here, not run.
# Argument selection for T537
The sole free parameter is `g`, a **discrete_index** identifying a finite,
simple, undirected, connected, unweighted graph with at least two vertices.
It names the object whose shortest-path distance characteristic polynomial is
stored. Graph order is derived from that graph. There is no independently
selected continuous parameter, weighting, temperature, root index or second
quantity. The polynomial variable `x` is an indeterminate, not an input column.
The default selection contains every connected isomorphism class on two through
six vertices. Counts by order are 1, 2, 6, 21 and 112, totaling 142. These are
verified against McKay's connected graph catalogues and the corresponding T237
and T238 keys. `selected-graphs.json` lists every identity explicitly; the
generator reads that file in order. Keys are graph6 encodings obtained with
`canonical_label(algorithm='sage')`. Canonicalization changes the representation,
not which isomorphism classes are included. Entries are ordered by vertex count
and then by graph6 string. Common graph names are copied from T238; comments
about Laplacian cospectrality are deliberately not copied because they describe
a different invariant.
This is a complete bounded census of a genuine discrete family, intended as a
reference for identifying polynomials from small shortest-path network and
spectral calculations. The cutoff is our practical, noncanonical choice, not a
mathematical threshold or a claim of a specific application for every graph.
It is independent of decimal notation. All trees, cyclic graphs, bipartite and
non-bipartite graphs, and regular and irregular graphs in the bound are included.
Breadth therefore comes before extensions of any one named family. The selection
includes C6 and K3,3, as well as graphs with more than one positive distance
eigenvalue. Keeping only complete graphs or paths would obscure these behaviors.
The one-vertex graph contributes only the polynomial x, which is stated in a
comment. Seven vertices would add 853 graphs. That would mostly deepen the census;
it is deferred pending a reason to extend this first reference range. The
measured longest stored polynomial is only 46 characters, so readability and
storage do not force this cutoff. The purpose and coverage judgment do. No
existing entries are being removed and there is no Cartesian product of inputs.
The choice agrees with family issue 217, but is justified anew under the archived
canonical skill. The monic convention det(xI-D), variable x, shortest-path edge
counts, graph6 backend and parameter identity all agree with that family. The
alternative historical convention det(D-xI) differs by (-1)^n and is explained
in the table; it does not require another parameter.
Checked supporting sources:
* Hogben and Reinhart, arXiv:2103.00647, Section 1 (distance matrix), Section 2.2
(the monic distance characteristic polynomial and the historical sign), and
Section 5 (complete, bipartite and cyclic families).
* McKay, https://users.cecs.anu.edu.au/~bdm/data/graphs.html, simple connected
graph counts and the downloaded graph2c.g6 through graph6c.g6 catalogues.
* McKay, https://users.cecs.anu.edu.au/~bdm/data/formats.html, graph6 encoding.
* Live T237 and T238, read 2026-10-27, for stable existing graph identities.
Policy evidence: agents/runs/20261027T040650Z-build-skills/manifest.json.
Both archived skill files have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
See the attached skill-evidence.json. This build justification is not the required
independent critique or acceptance. The controller must obtain that in its
separate critique session, fingerprinting generate.py and selected-graphs.json
(and any other source dependencies that a later revision introduces).