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3508 bytes, as of the version from 2026-10-27 04:15 (current). Recorded here, not run.
# Sources and conventions for T537
Read on 2026-10-27. Family proposal: numberdb-data issue 217, read through
`python3 agents/queue.py show 217` before the build. This provenance stays in
run evidence, not in the mathematical table prose.
* Hogben and Reinhart, https://arxiv.org/pdf/2103.00647, Section 1: graph
distance counts shortest-path edges. Section 2.2, p. 6: the distance
characteristic polynomial is det(xI-D); the older det(D-xI) convention has
sign (-1)^n. Section 2.1, Theorem 2.4: the tree distance determinant.
Section 5, Propositions 5.1 and 5.2: complete and complete-bipartite spectra.
The bipartite polynomial in the draft is also derived directly from its
two-part quotient matrix, extending without ambiguity to a=1 or b=1.
* Zhou and Ilic, https://arxiv.org/pdf/1101.4393, Introduction: confirms the
shortest-path convention and largest eigenvalue interpretation. Consulted
for family agreement; not added as an uncited bibliography entry.
* McKay, https://users.cecs.anu.edu.au/~bdm/data/graphs.html: connected graph
counts 1, 2, 6, 21, 112, and 853 for orders 2 through 7. Files graph2c.g6
through graph6c.g6 downloaded from that same directory are attached and
checked against every selected isomorphism class.
* McKay, https://users.cecs.anu.edu.au/~bdm/data/formats.html: graph6 encoding.
* T237, https://numberdb.org/Characteristic_polynomials_of_connected_graphs:
adjacency polynomial convention and common graph identities.
* T238, https://numberdb.org/Laplacian_polynomials_of_connected_graphs:
common graph identities and names. Non-name comments about Laplacian
cospectrality were not transferred.
* T330, https://numberdb.org/Characteristic_polynomials_of_the_classical_test_matrices:
its Fiedler definition is |i-j|. Every stored Fiedler polynomial for n=2..6
agrees with the corresponding path distance polynomial. This is a relation
between two object descriptions; values were independently generated.
* T536, https://numberdb.org/Distance_spectral_radii_of_connected_graphs:
same 142 graph identities; all largest roots were checked against its
100-digit intervals or exact integer values. The actual address was read
from the table's files page. T536 was a draft at the time of the build.
The coefficient of x^(n-2) is minus one half the trace of D^2, hence minus
sum d(u,v)^2 over unordered pairs. This follows directly by summing the
2-by-2 principal minors. The optional checker also verifies the analogous
third coefficient via unordered triples, the tree inertia, and constant
transmission roots. These additional checks are not extra table quantities.
Corpus screening used both supplied screen functions, title/definition
searches, and api/lookup on x^2-1, x^3-6x-4 and x^3-3x-2. Matches were to other
families (including adjacency and Fiedler polynomials), not a distance
characteristic-polynomial catalogue. already_asked returned no title match;
family 217 itself cites the earlier general graph-spectra proposal 187.
No request provenance was put in the table's mathematical prose.
Existing tags polynomial, combinatorics and eigenvalue were confirmed. The tag
`graph theory` was absent. It is proposed for this table together with T125,
T126, T237, T238, T239, T248, T397 and T400; no published table was edited.
Skill manifest: agents/runs/20261027T040650Z-build-skills/manifest.json.
Archived local and live instructions both have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.