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3909 bytes, as of the version from 2026-10-27 11:08 (current). Recorded here, not run.
# Argument selection for T545
The only free parameter is `g`, a graph isomorphism class. Its category is
`discrete_index`. Its domain is finite simple connected undirected unweighted
graphs with at least two vertices. Order, degree, adjacency entries and every
eigenvalue are derived from the graph; none is an additional free parameter.
The quantity is the adjacency Estrada index with coefficient one in the
exponential and no division by the number of vertices.
The default enumeration includes every isomorphism class of orders 2 through
6: 1, 2, 6, 21 and 112 classes, respectively, totaling 142. This is an exhaustive
small-order structural census. It permits recognition and comparison of the
closed-walk sums across branching trees, cycles, bipartite and non-bipartite
graphs, and regular and irregular graphs. These groups are all included without
pruning by the simplicity or fame of an individual answer. This is discrete
coverage, not a selection of continuous reference inputs or a claim of a
published application for every graph. It introduces no weights, temperatures,
edge probabilities or products of auxiliary parameter choices.
Six is a practical, noncanonical boundary, independent of a numeral base. It
provides those different structural families before increasing their depth.
Order seven alone would add 853 classes, about six times the current row count.
Keeping the first census small makes the graph identities and comparisons
reviewable; the 100-digit value length stays essentially constant with order,
so this cutoff is a choice about coverage and review effort, not a storage
limit. The dry run measured 142 entries and a 26.2 KB block, well within the
limits. An extension can reconsider the cutoff. Excluding the single-vertex
graph follows the common connected-graph family domain and omits the trivial
value 1. No larger named graphs have been appended.
Graph6 is an encoding, not a canonicalization rule. Identity is the graph6
string after `canonical_label(algorithm='sage')`, tested in SageMath 10.9.
Entries are ordered by vertex count, then ASCII graph6 order. Explicit
`Graph(key, format='graph6')` decoding preserves literal backslashes. The
canonical keys agree with all entries of orders 2 through 6 in T237, T238,
T397 and T400. McKay's downloaded catalogues provide an additional census
control, independently of the existing NumberDB keys. Names are attached only
for graphs constructed as complete graphs, paths, cycles or complete bipartite
graphs and matched by the same canonical identity.
Sources checked:
- E. Estrada, *The many facets of the Estrada indices of graphs and networks*,
SeMA Journal 79 (2022), 57–125, DOI 10.1007/s40324-021-00275-w:
section 2 fixes simple connected graph conventions; sections 3 and 5,
especially (5.1), relate the adjacency spectrum and closed walks.
https://link.springer.com/article/10.1007/s40324-021-00275-w
- Brendan McKay, *Graphs*, the connected simple graph counts at orders 2–7,
and the linked graph2c.g6 through graph6c.g6 files:
https://users.cecs.anu.edu.au/~bdm/data/graphs.html
- Brendan McKay, graph6 format specification:
https://users.cecs.anu.edu.au/~bdm/data/formats.html
- Live NumberDB T237, T238, T397 and T400, read through the API in this build;
`corpus-reference.json` archives the relevant entries. Their slugs were read
from search results, not derived from their titles.
Policy evidence: the runner's archived manifest is
`agents/runs/20261027T103803Z-build-skills/manifest.json`, also attached as
`skill-manifest.json`. Both local and live versions have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
`skill-evidence.json` records this build's read and run identifier. This file is
the builder's justification. The controller must obtain the separate critique
and its argument-review sidecar; this build does not supply that approval.