Two different graphs with the same spectrum, to the last digit

At its defaults it draws two different graphs with the same spectrum, to the last digit. a triangle and a square, joined and K₂,₃ with a pendant edge: 6 vertices and 7 edges each, drawn above their combinatorial Laplacian spectra. The two characteristic polynomials are the same integer polynomial, checked in exact arithmetic rather than by comparing computed eigenvalues, and the computed spectra differ by 5.33·10⁻¹⁵ — which is the eigensolver's own rounding and not a property of the graphs. Their degree sequences are 4, 2, 2, 2, 2, 2 and 3, 3, 3, 2, 2, 1, so they are not the same graph, and every method in this field that reads only a spectrum answers about the class rather than the graph.

cospectral-pair is one function in lib/figures/graphres.js — resistance, sparsification and elimination — three routes to one number, and a count that outgrows the format. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

a triangle and a square, joined and K₂,₃ with a pendant edge: 6 vertices and 7 edges each, drawn above their combinatorial Laplacian spectra. The two characteristic polynomials are the same integer polynomial, checked in exact arithmetic rather than by comparing computed eigenvalues, and the computed spectra differ by 5.33·10⁻¹⁵ — which is the eigensolver's own rounding and not a property of the graphs. Their degree sequences are 4, 2, 2, 2, 2, 2 and 3, 3, 3, 2, 2, 1, so they are not the same graph, and every method in this field that reads only a spectrum answers about the class rather than the graph.

Every pair of non-isomorphic connected graphs with the same Laplacian characteristic polynomial, as a share of all such pairs at each size. 6 vertices: 2 pairs, 1 separated by the critical group, 1 with both groups cyclic, 0 sharing one non-cyclic group; 7 vertices: 65 pairs, 30 separated by the critical group, 23 with both groups cyclic, 12 sharing one non-cyclic group; 8 vertices: 1022 pairs, 435 separated by the critical group, 361 with both groups cyclic, 226 sharing one non-cyclic group.

Left pair: ℤ₁₂ against ℤ₂ ⊕ ℤ₆, 12 spanning trees each, told apart by the group. Right pair: ℤ₂₄ for both, 24 spanning trees each, and a cyclic group is determined by its order, so the group cannot tell them apart; their degree sequences can.

The share of connected graphs whose grounded Laplacian has at most one invariant factor above one: 100.0% on 3 vertices, 83.3% on 4 vertices, 66.7% on 5 vertices, 61.6% on 6 vertices, 61.0% on 7 vertices, 63.6% on 8 vertices. Among graphs that have a Laplacian-cospectral mate: 75.0% on 6, 51.3% on 7, 48.5% on 8.

Every abelian group of squarefree order is cyclic, so a pair whose tree count is squarefree cannot be separated by its critical group. squarefree tree count: 157 pairs, 0 separated; a square divides it: 865 pairs, 435 separated.

Pairs of non-isomorphic connected graphs on 8 vertices with the same Laplacian characteristic polynomial, and how many each invariant separates: critical group 435 of 1022; degree sequence 354 of 1022; triangle count 352 of 1022; signless Laplacian spectrum 906 of 1022; adjacency spectrum 1022 of 1022; group or degree sequence 640 of 1022.

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

40 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

It draws a decomposition and prints its residual. It calls jacobiEigSym, spectrum, spectrum, spanningTreesExact, and every figure above carries the badge — which residualcheck verifies by looking for it in the emitted SVG rather than by finding the call that builds one. A badge that is constructed and then left out of the body is the failure that check exists for.

Across the library: the rule bites on 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.