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# Argument selection for Graph energies of connected graphs (T546)
## Policy and family
The governing instructions are the archived canonical local and live skills in
`agents/runs/20261027T111010Z-build-skills/manifest.json`. Both were read and their
SHA-256 is `e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The source snapshot's canonical file has the same digest. This is the builder's
justification, not the required independent critique or its acceptance sidecar.
Family issue numberdb-data#220 was read before work. Its common conventions
specify simple, connected, undirected, unweighted graphs with at least two
vertices. Graph energy is the sum of the absolute adjacency eigenvalues with
algebraic multiplicity, as defined in Das and Gutman, Section 1, equation (1).
Neither a Laplacian energy nor a normalized adjacency matrix is implicit.
## Every free parameter
The only free parameter is `g`, a `Symbolic` graph isomorphism class. Its role is
`discrete_index`. The graph order, edge count, degrees, adjacency matrix and
spectrum are derived from this object, not separately sampled arguments.
There are no continuous parameters or mixed selection groups.
The default enumeration contains every connected simple graph on 2, 3, 4, 5
and 6 vertices, once up to isomorphism: respectively 1, 2, 6, 21 and 112 graphs.
The total is 142. McKay's independently supplied catalogues have these counts,
and canonicalizing their contents gives exactly the selected set. The selection
is not inferred solely from an existing NumberDB table's size.
## Purpose, breadth and practical boundary
This is an exhaustive small-order structural census for recognizing graph
energies, including values from irregular topologies and from several positive
and negative spectral modes. Restricting to complete and complete-bipartite
graphs would overrepresent energies reducible to elementary multiples of one
spectral radius. The full census contains 13 trees, 129 graphs with cycles,
27 bipartite and 115 non-bipartite graphs, and 11 regular and 131 irregular
graphs. These classes overlap and are coverage diagnostics, not additional
selection parameters. Five complete graphs, nine unordered complete-bipartite
part-size pairs, five paths and four cycles provide recognizable controls.
No separate application is claimed for each graph. The family is a genuine
discrete object domain; the bound of six is a practical, noncanonical boundary,
independent of decimal notation. All the structural groups just described are
already present by six vertices. Order seven adds 853 graphs, more than six
times the current rows, so extending breadth across small topologies is preferred
to adding that next order here. There are no silently appended larger named
graphs, arbitrary numerical grids or products of free inputs. No graph is
removed because its energy is rational, repeated, or unfamiliar.
The one-vertex graph is excluded to agree with the family's domain, which is
also used for algebraic connectivity. Its energy would be zero. The definition
of `g` has no upper-order bound; the actual 2-through-6 coverage belongs in the
completeness note. Enumeration orders rows by vertex count and then ASCII
order of the canonical graph6 string. SageMath 10.9 with
`canonical_label(algorithm='sage')` supplies the identities. Every selected
identity agrees with the order-2-through-6 subset of T237, T238, T397 and T400,
and with the current T545 census. Graph6 is an encoding, not the canonicalization
algorithm. Decoding is explicit and preserves literal backslashes.
## Precision and exactness
There are 13 provably integral energies and 129 irrational energies. Integer
characteristic polynomials are factored over the rationals and their real roots
isolated exactly. When every root of an irreducible factor has one sign, its
contribution is its signed trace, an integer, times its multiplicity. Mixed-sign
factors contribute certified intervals; no algebraic compositum is constructed.
This criterion does not miss a rational total through cancellation between
factors. In the totally real splitting field, write the energy as the sum of
each eigenvalue times its original sign. Every Galois conjugate of that signed
sum is at most the original energy, term by term, since the conjugates permute
the spectrum. If a mixed-sign irreducible factor occurs, transitivity gives an
automorphism taking a positive root to a negative one. For that automorphism
the inequality is strict. The energy therefore cannot be rational. With no
mixed-sign factors the trace computation is exact. This establishes exactness
from the definition rather than from a run of numerical zeros.
At 413 working bits, every nonexact interval has width at most
`3/1322111937580497197903830616065542079656809365928562438569297590548811582472622691650378420879430569695182424050046716608512`
(less than 2.27e-123). All written nonexact entries carry 100 significant digits;
the full certified interval lies inside the written interval. The longest
written number is 101 characters, and the flat JSON entries block is 19,777
bytes. The generic dry-run helper measures the working-precision representation
as 126 characters at most and its entries block as 21.8 KB. These measured sizes
support readability; they did not determine the mathematical selection.
## Checked sources
- Kinkar Ch. Das and Ivan Gutman, *Bounds for the energy of graphs*, Hacettepe
Journal of Mathematics and Statistics 45(3) (2016), 695–703. Section 1,
equation (1) fixes energy, adjacency and multiplicity; Section 3, equation (3)
gives the complete-bipartite lower-bound equality.
https://scidar.kg.ac.rs/bitstream/123456789/9133/1/10.15672-HJMS.20164513097.pdf
The journal metadata was checked at
https://dergipark.org.tr/en/pub/hujms/article/524354 .
- Brendan McKay, simple connected graph catalogues, orders 2 through 7:
https://users.cecs.anu.edu.au/~bdm/data/graphs.html . The order-2-through-6
files were downloaded and compared as sets, not only counted.
- Brendan McKay, graph6 format specification, upper-triangle bit ordering:
https://users.cecs.anu.edu.au/~bdm/data/formats.txt . Its DQc example is a
control for the independent decoder.
- SageMath reference, Algebraic numbers, interval conversion:
https://doc.sagemath.org/html/en/reference/number_fields/sage/rings/qqbar.html .
- NumberDB T237, T238, T397, T400 and T545, read from the API. T237 also supplies
independent stored integer characteristic polynomials. Their existence is
evidence for shared identifiers, not by itself evidence for this cutoff.
## Independent checks and remaining review
`checks.py` decodes graph6 independently, computes exact closed-walk traces and
uses Newton identities to reconstruct every characteristic polynomial. It also
diagonalizes all 142 integer adjacency matrices with mpmath at 150 and 180
decimal digits, independently of Sage's polynomial/root path. The two numerical
precisions agree to much better than 140 decimal places. Known controls are
run first. Complete, complete-bipartite, path and cycle formulas, trace identities
and energy bounds are checked for every applicable row. These floating-point
checks provide independent agreement, not the certification; the latter comes
from the exact root isolation and interval arithmetic.
The separate critique must assess this argument, the actual stored enumeration,
the generator sources and the numerical claims, then write its own fingerprinted
sidecar. This build neither creates that acceptance nor offers the draft.