An integer elliptic Sombor energy

Abstract

Two four-cycles sharing a vertex have elliptic Sombor energy 144.

Definition 1.1 (The elliptic Sombor matrix).

$$\forall (n:\mathbb{N}),\forall (G:\mathrm{SimpleGraph}\left(\mathrm{Fin}\left(n\right)\right)),[\mathrm{DecidableRel}\left(\mathrm{Adj}\left(G\right)\right)],\forall (i:\mathrm{Fin}\left(n\right)),\forall (j:\mathrm{Fin}\left(n\right)),\mathrm{ellipticSomborMatrix}\left(G,i,j\right)=\{\begin{array}{rl}((\mathrm{val}\left(\mathrm{degree}\left(G,i\right)\right):\mathbb{R})+(\mathrm{val}\left(\mathrm{degree}\left(G,j\right)\right):\mathbb{R}))\cdot \sqrt{(\mathrm{val}\left(\mathrm{degree}\left(G,i\right)\right):\mathbb{R}{)}^2+(\mathrm{val}\left(\mathrm{degree}\left(G,j\right)\right):\mathbb{R}{)}^2}& \text{if}\mathrm{Adj}\left(G,i,j\right)\\ 0& \text{otherwise}\end{array}$$

Formalization. D5/S3/Combinatorics/Graph/EllipticSomborEnergyIntegerRefutation.ellipticSomborMatrix (✓ std3).

Citation. Saeid Alikhani, Nima Ghanbari, Mohammad Ali Dehghanizadeh (2024). Elliptic Sombor energy of a graph. DOI: 10.48550/arXiv.2404.18622. URL: https://arxiv.org/abs/2404.18622v1.

Commentary.

Abstract, page 1: “Let G be a simple graph with vertex set V(G) = {v₁, v₂, …, vₙ}. The elliptic Sombor matrix of G, denoted by A_ESO(G), is defined as the n×n matrix whose (i,j)-entry is (dᵢ+dⱼ)√(dᵢ²+dⱼ²) if vᵢ and vⱼ are adjacent and 0 for another cases.” Vertices are labelled by Fin n, starting at zero. SimpleGraph.degree counts adjacent vertices. Typed val denotes the natural-number degree cast into ℝ before addition and squaring. Each entry is zero on a nonadjacent pair, including the diagonal.

Definition 1.2 (Conjecture 3.9).

$$claim\iff (\forall (n:\mathbb{N}),\forall (G:\mathrm{SimpleGraph}\left(\mathrm{Fin}\left(n\right)\right)),[\mathrm{DecidableRel}\left(\mathrm{Adj}\left(G\right)\right)],\forall (hA:\mathrm{IsHermitian}\left(\mathrm{ellipticSomborMatrix}\left(G\right)\right)),\forall (z:\mathbb{Z}),\sum _{i:\mathrm{Fin}\left(n\right)}|\mathrm{eigenvalues}\left(hA,i\right)|\ne (\mathrm{val}\left(z\right):\mathbb{R}))$$

Formalization. D5/S3/Combinatorics/Graph/EllipticSomborEnergyIntegerRefutation.claim (✓ std3).

Citation. Saeid Alikhani, Nima Ghanbari, Mohammad Ali Dehghanizadeh (2024). Elliptic Sombor energy of a graph. DOI: 10.48550/arXiv.2404.18622. URL: https://arxiv.org/abs/2404.18622v1.

Commentary.

Conjecture 3.9, page 12: “There is no graph with integer-valued elliptic Sombor energy.” Abstract, page 1: “The elliptic Sombor energy E_ESO of G is the sum of absolute values of the eigenvalues of A_ESO(G).” Encoding: n is a natural number, G is any simple graph on Fin n, and adjacency is decidable. The matrix is always Hermitian because adjacency is symmetric. hA is a proof of that property and supplies Mathlib’s eigenvalues, indexed by Fin n with algebraic multiplicity. The sum is independent of the choice of Hermitian proof. Every integer z is cast to ℝ; the claim excludes equality to all such casts. The empty graph is included in this literal all-graphs statement; the counterexample below has seven vertices and no isolated vertex.

Theorem 1.3 (Two squares with a common vertex).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Graph/EllipticSomborEnergyIntegerRefutation.result (✓ std3). ∎

Resolves. Problems/alikhani-ghanbari-dehghanizadeh-2024-elliptic-sombor-energy-integer-refutation (refuted) by D5/S3/Combinatorics/Graph/EllipticSomborEnergyIntegerRefutation.result.

Source. Repository-derived.

Acknowledgement. Saeid Alikhani, Nima Ghanbari, Mohammad Ali Dehghanizadeh (2024). Elliptic Sombor energy of a graph. DOI: 10.48550/arXiv.2404.18622. URL: https://arxiv.org/abs/2404.18622v1.

Commentary.

Use the edges 0–1–2–3–0 and 0–4–5–6–0. Vertex zero has degree four and every other vertex has degree two. The four edges incident to zero have weight 12√5, and the other four have weight 8√2. An explicit invertible change of basis diagonalizes this real symmetric matrix with diagonal entries −56, −16, 0, 0, 0, 16, 56. Its characteristic polynomial is x³(x−56)(x+56)(x−16)(x+16). Mathlib’s spectral theorem identifies the roots with the Hermitian eigenvalue multiset. The absolute values therefore sum to 144, an integer, so the conjecture is false.

References

• Truth anchor: D5/S3/Combinatorics/Graph/EllipticSomborEnergyIntegerRefutation.claim
• Truth anchor: D5/S3/Combinatorics/Graph/EllipticSomborEnergyIntegerRefutation.ellipticSomborMatrix
• Truth anchor: D5/S3/Combinatorics/Graph/EllipticSomborEnergyIntegerRefutation.result