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4506 bytes, as of the version from 2026-10-27 03:45 (current). Recorded here, not run.
# Sources and conventions for T536
## Definition and graph families
Bo Zhou and Aleksandar Ilić, *On distance spectral radius and distance energy
of graphs* (2011), https://arxiv.org/abs/1101.4393. Introduction, page 2:
distances count shortest-path edges; the radius is the largest distance-matrix
eigenvalue. Section 2, page 3: the distance matrix of a graph of diameter at
most two is 2J-2I-A; for a regular graph its largest eigenvalue is 2n-2-r.
These support the definition and regular-graph check. The reference identifies
a real mathematical family, rather than merely a library function.
Leslie Hogben and Carolyn Reinhart, *Spectra of variants of distance matrices
of graphs and digraphs: a survey* (2021),
https://arxiv.org/abs/2103.00647, version 2. Section 1 fixes the connected,
undirected setting. Theorem 2.4 supplies the tree distance determinant and
inertia. Section 5, Propositions 5.1 and 5.2, supplies complete and complete
bipartite spectra; the subsequent cycle formula supplies its largest root.
The numerical checks test every applicable graph in the selected census.
The theorem's displayed tree hypothesis has a typographical omission in the
PDF, but its surrounding paragraph and notation identify trees explicitly.
The complete bipartite radius formula also covers a part of size one.
Indeed the positive Perron eigenvector is constant on each part and the
quotient matrix is [[2(a-1), b], [a, 2(b-1)]]. Its larger root is
(a+b-2)+sqrt(a*a-a*b+b*b), valid for all a,b>=1. This direct derivation covers
stars and K2, which lie outside Proposition 5.2's stated a,b>=2 range.
For a cycle, the row sum is floor(n*n/4); for a complete graph it is n-1.
Constant row sums and nonnegativity make these row sums the spectral radius.
## Enumeration and identifiers
Brendan McKay, *Graphs*, Simple graphs section,
https://users.cecs.anu.edu.au/~bdm/data/graphs.html. Counts of connected graphs
on orders 2..6 are 1,2,6,21,112; order 7 has 853. The downloaded graphNc.g6
files are archived beside these notes, and their URLs and SHA-256 digests are
in source-files.json. The catalogue strings are canonicalized with Sage's
explicit backend before comparison with the table's identities. They need
not match Sage keys before canonicalization.
McKay's graph6 definition:
https://users.cecs.anu.edu.au/~bdm/data/formats.html and
https://users.cecs.anu.edu.au/~bdm/data/formats.txt. The checker decodes its
upper-triangle bit ordering directly, independently of Sage's Graph parser.
T237 https://numberdb.org/Characteristic_polynomials_of_connected_graphs and
T238 https://numberdb.org/Laplacian_polynomials_of_connected_graphs supplied
the stable identities; the 142-element subsets agree. The graph names copied
from T237 are annotations, not an additional selection criterion.
## Existing numerical neighbors
T397 https://numberdb.org/Spectral_radii_of_connected_graphs stores adjacency
radii, a different invariant. T329
https://numberdb.org/Eigenvalues_of_the_classical_symmetric_test_matrices
includes Fiedler matrices with entries |i-j|. Ordering the vertices of a path
along the path identifies its distance matrix with this matrix. The checker
verifies this identity for all five paths and separately compares their
radii with T329's stored largest Fiedler eigenvalues. No values are imported
from either table. The connection to T329 is stated in the table and retained
as a mathematical equality of constructions rather than a copied data source.
## Independent spectral bounds
For every strictly positive vector v and irreducible nonnegative matrix D,
min_i (Dv)_i/v_i <= rho(D) <= max_i (Dv)_i/v_i. One proof multiplies the two
componentwise inequalities by a positive left Perron eigenvector. Our D has
strictly positive off-diagonal entries, so it is irreducible. Starting from
the all-ones vector, the checker repeatedly multiplies using integer
arithmetic, then forms exact rational bounds. It stops only when their
relative width is at most 10^-115. K2 and K3 are exact controls, and P3 has
the independently derived radius 1+sqrt(3). The written decimal interval is
required to contain the independent bounds, not just a close midpoint.
## Policy provenance
The local and live archived skill files both hash to
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
Manifest: `agents/runs/20261027T033406Z-build-skills/manifest.json`.
This run leaves the table unpublished and unoffered; independent critique
and controller actions are separate.