Two four-cycles sharing a vertex refute elliptic Sombor Conjecture 3.9

Alikhani, Ghanbari and Dehghanizadeh, Elliptic Sombor energy of a graph, arXiv:2404.18622v1, Conjecture 3.9, states:

There is no graph with integer-valued elliptic Sombor energy.

For a simple graph, the elliptic Sombor matrix has entry (d_i + d_j) sqrt(d_i^2 + d_j^2) on an adjacent pair and zero otherwise; the elliptic Sombor energy is the sum of the absolute values of its eigenvalues.

The conjecture is a universal nonintegrality assertion. Two copies of the four-cycle sharing one vertex give a seven-vertex simple connected bipartite graph with hub degree four and all other degrees two. Its elliptic Sombor matrix has spectrum {-56, -16, 0, 0, 0, 16, 56}, so its energy is 144.

The source conjecture quantifies over every finite simple graph. A refutation requires one literal graph, its literal degree-weighted matrix and a certified integer spectral energy. The source does not settle whether integer-energy graphs must be bipartite or classify the remaining graphs.

ellipticSomborMatrix is the source matrix on Fin n. claim is the literal universal nonintegrality statement, with Mathlib's Hermitian eigenvalue indexing. result : ¬ claim constructs the two-cycle bouquet, computes its matrix, diagonalizes it by an explicit invertible change of basis, identifies the characteristic roots and sums their absolute values to 144.

The refutation would fail if the bouquet were not a simple graph, if any edge weight or degree were wrong, if the change-of-basis matrices were not inverse, if the characteristic polynomial did not have roots -56, -16, 0, 0, 0, 16, 56, or if the Hermitian eigenvalue sum did not equal 144. Each condition is checked in the kernel-compiled result theorem.

The frozen declaration D5/S3/Combinatorics/Graph/EllipticSomborEnergyIntegerRefutation.result has type ¬ claim and uses the standard axiom closure {propext, Classical.choice, Quot.sound}. The public surface consists of the literal matrix definition, the source claim and the refutation result; the bouquet and matrix-basis calculations are private. The source is attested by Saeid Alikhani, Nima Ghanbari and Mohammad Ali Dehghanizadeh (2024), DOI 10.48550/arXiv.2404.18622, Conjecture 3.9.

What the settlement shows

Proved in this module: the two-copy bouquet B_2 has elliptic Sombor energy 144, so Conjecture 3.9 is refuted. The failure mechanism is the integral spectrum of this connected bipartite graph. The source paper's separately computed formulas for its listed graph classes survive; only the universal nonintegrality conjecture is removed. Whether every graph with integer elliptic Sombor energy is bipartite is open.

Computed: exact characteristic polynomials, spectra and energies for every B_k, k = 1..10, using Python 3 and SymPy 1.14.0 (exit 0). B_k consists of k copies of C₄ sharing vertex zero and has 3k + 1 vertices. The script constructs the degree-weighted matrix and computes its SymPy characteristic polynomial before extracting all roots with algebraic multiplicity. Energy and integrality use exact symbolic arithmetic.

In the spectrum column, [m] gives the multiplicity of each displayed value; ±a [m] means that each of −a and a has multiplicity m. Values without a bracket have multiplicity one.

| k | Characteristic polynomial (computed) | Spectrum (computed) | Energy (computed) | Integer? (computed) | | ---: | --- | --- | --- | --- | | 1 | $x^2(x^2-512)$ | $0\ [2],\ \pm16\sqrt{2}$ | $32\sqrt{2}$ | no | | 2 | $x^3(x^2-256)(x^2-3136)$ | $0\ [3],\ \pm16,\ \pm56$ | $144$ | yes | | 3 | $x^4(x^2-256)^2(x^2-15616)$ | $0\ [4],\ \pm16\ [2],\ \pm16\sqrt{61}$ | $64+32\sqrt{61}$ | no | | 4 | $x^5(x^2-256)^3(x^2-54656)$ | $0\ [5],\ \pm16\ [3],\ \pm8\sqrt{854}$ | $96+16\sqrt{854}$ | no | | 5 | $x^6(x^2-256)^4(x^2-150016)$ | $0\ [6],\ \pm16\ [4],\ \pm16\sqrt{586}$ | $128+32\sqrt{586}$ | no | | 6 | $x^7(x^2-256)^5(x^2-348352)$ | $0\ [7],\ \pm16\ [5],\ \pm8\sqrt{5443}$ | $160+16\sqrt{5443}$ | no | | 7 | $x^8(x^2-256)^6(x^2-717056)$ | $0\ [8],\ \pm16\ [6],\ \pm16\sqrt{2801}$ | $192+32\sqrt{2801}$ | no | | 8 | $x^9(x^2-256)^7(x^2-1348096)$ | $0\ [9],\ \pm16\ [7],\ \pm16\sqrt{5266}$ | $224+32\sqrt{5266}$ | no | | 9 | $x^{10}(x^2-256)^8(x^2-2361856)$ | $0\ [10],\ \pm16\ [8],\ \pm16\sqrt{9226}$ | $256+32\sqrt{9226}$ | no | | 10 | $x^{11}(x^2-256)^9(x^2-3910976)$ | $0\ [11],\ \pm16\ [9],\ \pm8\sqrt{61109}$ | $288+16\sqrt{61109}$ | no |

Only B_2 is integral in this computed range. A uniform statement for all other k is open.

The same computation applied to the ordinary Sombor matrix gives energies 8√2, 8 + 8√6, 48, and 24 + 8√35 for k = 1, 2, 3, 4. Thus B_3 gives the separately preregistered Ghanbari Conjecture 3.8 (refuted in issue #13388); this dossier does not settle any further Sombor statement. A uniform extension of the ordinary Sombor readings beyond the tested four values is open.

The computation script source is below. Save the block as bouquet-triage.py and run python3 bouquet-triage.py with SymPy 1.14.0 (python3 -m pip install sympy==1.14.0). The SHA-256 of its UTF-8 bytes, with LF line endings and one final newline, is da792c23330de23a81685943c3c4b39caec7e9390e109f77d76f656e2251164e.

import json
import sympy as sp

x = sp.Symbol("x")


def bouquet(k):
    edges = set()
    for r in range(k):
        a, b, c = 3 * r + 1, 3 * r + 2, 3 * r + 3
        edges.update({(0, a), (a, b), (b, c), (c, 0)})
    degrees = [0] * (3 * k + 1)
    for i, j in edges:
        degrees[i] += 1
        degrees[j] += 1
    return edges, degrees


def reading(k, elliptic):
    edges, degrees = bouquet(k)
    matrix = sp.zeros(len(degrees))
    for i, j in edges:
        value = sp.sqrt(degrees[i] ** 2 + degrees[j] ** 2)
        if elliptic:
            value *= degrees[i] + degrees[j]
        matrix[i, j] = matrix[j, i] = value
    polynomial = sp.Poly(sp.expand(matrix.charpoly(x).as_expr()), x)
    roots = sp.roots(polynomial.as_expr(), x)
    assert sum(roots.values()) == matrix.rows
    assert sp.Poly(sp.prod((x - value) ** count for value, count in roots.items()).expand(), x) == polynomial
    assert all(value.is_real is True for value in roots)
    energy = sp.simplify(sum(sp.Abs(value) * count for value, count in roots.items()))
    assert energy.is_integer in (True, False)
    return {
        "k": k,
        "vertices": matrix.rows,
        "characteristic_polynomial": str(sp.factor(polynomial.as_expr())),
        "spectrum": [{"eigenvalue": str(value), "multiplicity": int(count)} for value, count in sorted(roots.items(), key=lambda item: sp.default_sort_key(item[0]))],
        "energy": str(energy),
        "integer": bool(energy.is_integer),
    }


print(json.dumps({
    "sympy_version": sp.__version__,
    "elliptic": [reading(k, True) for k in range(1, 11)],
    "ordinary_sombor": [reading(k, False) for k in range(1, 5)],
}, indent=2))

Open: whether every graph with integer elliptic Sombor energy is bipartite, and whether the bouquet family has further integral values beyond the tested scope.

The literature check covers the cited arXiv version and the registered source note; it does not establish exhaustive worldwide priority or exclude an unindexed later settlement. The finite B_k computations above are symbolic SymPy readings, while the B_2 refutation itself is kernel-checked.