Ghanbari's integer Sombor energy conjecture

N. Ghanbari, On the Sombor characteristic polynomial and Sombor energy of a graph, Computational and Applied Mathematics 41 (2022), article 242, Conjecture 3.8:

There is no graph with integer-valued Sombor energy.

The paper studies finite simple graphs. For a graph $G$, the Sombor matrix has entry $\sqrt{d_i^2+d_j^2}$ on an adjacent pair and zero otherwise, and the Sombor energy is the sum of the absolute values of its eigenvalues.

The frozen declaration D5/S3/Combinatorics/Graph/SomborEnergyIntegerRefutation.result refutes the universal conjecture with three copies of $C_4$ sharing one vertex. The witness is connected, bipartite, has ten vertices and twelve edges, and has Sombor energy $48$.

Preregistration issue #13388 records the source quotation, the finite-simple-graph scope, the fully quantified Lean claim, the literature search, and the expected $C_4$ bouquet witness. The paper asks either for a proof of Conjecture 3.8 or a graph with integer-valued Sombor energy. The delivered result supplies the latter. The recorded searches are bounded and do not establish worldwide novelty or publication priority.

Represent the graph on $\operatorname{Fin} 10$ by three four-cycles $0-1-2-3-0$, $0-4-5-6-0$, and $0-7-8-9-0$. Vertex $0$ has degree six and every other vertex has degree two, so the six edges incident with $0$ have weight $\sqrt{40}$ and the other six edges have weight $\sqrt{8}$. The Lean proof writes an explicit matrix $P$, proves $P^\mathsf{T}P$ is diagonal, constructs its inverse $Q$, and proves the similarity identity with diagonal spectrum $(16,-16,4,-4,4,-4,0,0,0,0)$. The Hermitian characteristic-polynomial theorem then identifies the eigenvalue multiset, and the absolute-value sum is $48$.

A failure of the finite matrix identities, the characteristic-polynomial factorisation, or the energy calculation would falsify the delivered witness. A graph with a different integer energy would be a further witness but is not needed for this refutation.

The canonical source is D5/S3/Combinatorics/Graph/SomborEnergyIntegerRefutation.lean. Its public surface is somborMatrix, claim, and result; the private proof contains the graph, degree calculation, matrix certificate, characteristic polynomial, and energy calculation. The Scribe mirror describes all three declarations and attaches OpenProblemResolutionClaim(Refuted) with this dossier's slug. The axiom closure of every public declaration is contained in {propext, Classical.choice, Quot.sound}.

The cited library note is Library/GraphInvariants/ghanbari2022somborenergy.md; its DOI and arXiv locator are verified.

Tier 1 external named conjecture, preregistered in #13388. Resolution Refuted by the frozen result. The module is proof_shape: bind-only, with escape_witness: none and admission_basis: open-problem-resolution (#13388; Refuted). Its utility is a certified instance refuting the module's claim. Information-escape registration is paused under CLAUDE.md section 3.9.

What the settlement shows

• Proved in this module: the bouquet of three copies of $C_4$ has Sombor spectrum $(16,-16,4,-4,4,-4,0,0,0,0)$ and energy $48$, so Ghanbari's universal Conjecture 3.8 is false. The failure mechanism is an explicit connected bipartite simple graph with a degree-six shared vertex; the source's finite-simple-graph hypotheses all hold.

• Computed for $1\leq k\leq 10$: for $B_k$, the bouquet of $k$ four-cycles sharing one vertex, the script below constructs the weighted Sombor matrix, factors its characteristic polynomial, and computes the exact energy. The energies are $8\sqrt{2}$, $8+8\sqrt{6}$, $48$, $24+8\sqrt{35}$, $32+8\sqrt{66}$, $40+32\sqrt{7}$, $48+32\sqrt{11}$, $56+24\sqrt{29}$, $64+8\sqrt{370}$, and $72+8\sqrt{506}$, respectively. Within this tested scope, $k=3$ is the only integral value. Any uniform statement for all $k$ is open. Command: python3 /tmp/sombor_bouquet_scan.py; exit code 0. Script SHA-256: 92f13a47c7e63f4e91e9573a4ef9ccb98591a3cd5008a38e7be04a95343efc89. The exact script source is:

  import sympy as sp
  
  x = sp.symbols('x')
  
  def bouquet_matrix(k):
      n = 3*k + 1
      M = sp.zeros(n)
      a = sp.sqrt((2*k)**2 + 2**2)
      b = sp.sqrt(2**2 + 2**2)
      for t in range(k):
          u, v, w = 1 + 3*t, 2 + 3*t, 3 + 3*t
          for i, j, q in ((0, u, a), (u, v, b), (v, w, b), (w, 0, a)):
              M[i, j] = q
              M[j, i] = q
      return M
  
  for k in range(1, 11):
      M = bouquet_matrix(k)
      poly = sp.factor(M.charpoly(x).as_expr(), extension=[sp.sqrt(2), sp.sqrt(k*k+1)])
      vals = M.eigenvals()
      energy = sp.simplify(sum(mult * abs(sp.N(val, 50)) for val, mult in vals.items()))
      exact_terms = []
      for val, mult in vals.items():
          if val.is_real and val.is_positive:
              exact_terms.append(mult * val)
      exact = sp.simplify(2 * sum(exact_terms))
      print(f"k={k} vertices={3*k+1} charpoly={poly} energy={exact} numeric={sp.N(exact, 18)}")
  

• Open: the separate regular-graph conjecture discussed by Ramane–Kitturmath (DOI: 10.1016/j.exco.2023.100115) is not the universal Conjecture 3.8 settled here and receives no claim in this module.

• Open: the minimum vertex count of a finite simple graph with integer Sombor energy is not established by this delivery. The computed $B_k$ table is only the tested range $1\leq k\leq 10$.

• Surviving source boundary: the source's characteristic-polynomial and energy calculations for graph classes other than the refuted universal claim are unaffected; only the universal no-integer-energy conjecture is refuted.

The literature search and the absence of an earlier published counterexample are bounded by the checks recorded in #13388. No claim is made about worldwide priority. A uniform classification of all $B_k$, the regular-graph conjecture, and the minimum-vertex problem remain open here.