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# Argument selection for T538
The only free parameter is `g`, a Symbolic identity of a finite, simple,
undirected, connected, unweighted graph on at least two vertices. Its role is
**discrete_index**: the underlying argument is an isomorphism class of genuine
finite graphs. Graph6 is an address, not a continuous parameter or a numerical
mesh. The distance Estrada index is one quantity, with no temperature, edge
weight, normalization, or spectral-index parameter.
We select every connected isomorphism class with 2 through 6 vertices. There
are 1, 2, 6, 21, and 112 classes by order, hence 142 entries. This is an
exhaustive bounded census within a discrete family. The finite six-vertex
boundary is a practical and noncanonical reference choice. It is independent
of numerical notation and does not claim an observed application for each
individual graph. The goal is to identify values obtained in small-network
shortest-path spectral calculations and matrix-exponential experiments.
Breadth is built into the census: it contains all small trees, cyclic graphs,
bipartite and non-bipartite graphs, and regular and irregular graphs. Including
all graphs through six vertices reaches C6 and K3,3 and includes spectra with
zero eigenvalues and more than one positive eigenvalue. Such cases test the
full exponential sum beyond the single dominant eigenvalue or tree identities.
Keeping only complete graphs, paths, and stars would remove most small-network
reference inputs. Seven vertices would add 853 isomorphism classes; that
expansion mainly increases the census, whereas six vertices already includes
the structural cases named here. Its benefit should be reassessed before an
extension. The one-vertex graph has index exactly 1 and is stated in prose.
The same selection as the distance radius and characteristic-polynomial tables
is justified by their spectral relationship: one gives the largest eigenvalue,
one encodes all eigenvalues, and this table gives their exponential sum. It is
not a mesh reused across unrelated continuous functions. No mixed groups or
Cartesian products are hidden in the selection; vertex count is derived from
the graph and is not a second free parameter.
`selected-graphs.json` contains the whole ordered selection and the 26 common
names retained from T237. Two adjacency-cospectral annotations were not copied,
since they did not name the graphs and concerned another matrix. Canonical keys
are Sage `canonical_label(algorithm='sage').graph6_string()`. They agree with
the relevant T237 and T238 keys; every key is checked against an independent
nauty enumeration. Entry order is vertex count followed by graph6 string.
The default generator enumeration and the completeness note use this range.
Sources checked:
- Güngör and Bozkurt, *On the Distance Estrada Index of Graphs*, Hacettepe J.
Math. Stat. 38(3) (2009), 277–283, introduction and Definition 2.1, equation
(8): the connected simple graph, shortest-path distance matrix, and
uncentered sum of exponentials of signed eigenvalues. Equation (9) gives
the spectral-moment series.
https://dergipark.org.tr/en/download/article-file/86791
- Brendan McKay, *Combinatorial Data: Graphs*, “Simple graphs”: connected graph
counts for orders 2–7 and independent catalogues.
https://users.cecs.anu.edu.au/~bdm/data/graphs.html
- Brendan McKay, *Description of graph6, sparse6 and digraph6 encodings*,
graph6 section: the graph representation and edge ordering.
https://users.cecs.anu.edu.au/~bdm/data/formats.txt
Policy evidence: `agents/runs/20261027T044107Z-build-skills/manifest.json`.
Both archived skill files were read and hashed SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The attachment `skill-evidence.json` records both versions. This build's
justification is not an independent argument-selection acceptance; the
controller must obtain that in its separate critique session.