Locally Equienergetic Graphs
Pith reviewed 2026-05-10 11:13 UTC · model grok-4.3
The pith
Some pairs of graphs with the same number of vertices share identical summed local energies even when their structures differ.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Two graphs G and H of the same order are locally equienergetic if the sum over all vertices v of (energy of G minus energy of G minus v) equals the corresponding sum for H.
What carries the argument
The local energy invariant e(G), obtained by summing the energy differences that result from deleting each vertex one at a time.
If this is right
- Non-isomorphic graphs of equal order can still be indistinguishable by this summed local-energy measure.
- The new invariant supplies an additional test that graphs must pass before they can be declared non-equivalent under local energy.
- Constructions that preserve the sum of vertex-deletion energy differences produce further locally equienergetic pairs.
- Standard equienergetic graphs may or may not remain equienergetic once the local version of the measure is applied.
Where Pith is reading between the lines
- The invariant might be useful for distinguishing graphs in reconstruction problems where only deletion effects are observed.
- It could be checked whether local energy correlates with other vertex-centrality measures such as degree or betweenness.
- One could ask whether the equality of local energies forces equality of the underlying energy spectra or adjacency eigenvalues.
Load-bearing premise
That subtracting the energy of the graph after one vertex is deleted and then adding those differences across all vertices yields a stable and comparable quantity for different graphs.
What would settle it
Direct computation, for any specific pair presented in the paper, of the two summed local-energy values; the values must match exactly for the pair to be locally equienergetic.
read the original abstract
For a given graph \( G \), let \( G^{(j)} \) denote the graph obtained by the deletion of vertex \( v_j \) from \( G \). The difference \( \mathscr{E}(G) - \mathscr{E}(G^{(j)}) \) quantifies the change in the energy of \( G \) upon the removal of \( v_j \), termed as the local energy of \( G \) at vertex $v_j$, as defined by Espinal and Rada in 2024. The local energy of $G$ at vertex $v$ is denoted by \(\mathscr{E}_G(v)\). The local energy of the graph \( G \), therefore, is the summation of these vertex-specific local energies across all vertices in \( V(G) \), expressed by \( e(G) = \sum \mathscr{E}_G(v) \). Two graphs of the same order are defined as locally equienergetic if they have identical local energy. In this paper, we have investigated several pairs of locally equienergetic graphs.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript recalls the 2024 definition of local energy at a vertex v as ℰ(G) − ℰ(G − v) and defines the graph invariant e(G) as the sum of these quantities over all vertices. It then states that pairs of graphs of equal order having identical values of e(G) are locally equienergetic and reports that several such pairs have been investigated.
Significance. The definition of e(G) is unambiguous and extends the classical notion of equienergetic graphs by incorporating vertex-deletion effects. Concrete, verified examples of non-isomorphic pairs with equal e(G) would furnish new data points for spectral graph theory and could motivate further study of this invariant. The current text, however, supplies neither the pairs nor any verification, so the potential contribution remains unrealized.
major comments (1)
- Abstract: the central claim that 'several pairs of locally equienergetic graphs' have been investigated is unsupported by any explicit list of graphs, adjacency matrices, energy values, or computational verification, so the claim cannot be checked or reproduced from the manuscript.
Simulated Author's Rebuttal
We thank the referee for their detailed review and constructive feedback on our manuscript. We agree that the current presentation does not adequately support the claim of having investigated several pairs, and we will revise accordingly to improve clarity and reproducibility.
read point-by-point responses
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Referee: Abstract: the central claim that 'several pairs of locally equienergetic graphs' have been investigated is unsupported by any explicit list of graphs, adjacency matrices, energy values, or computational verification, so the claim cannot be checked or reproduced from the manuscript.
Authors: We acknowledge that the manuscript states we have investigated several pairs of locally equienergetic graphs but does not include the specific examples, graph descriptions, computed values of e(G), or verification steps. This was an oversight in the drafting process. In the revised version, we will add a dedicated section presenting at least three concrete non-isomorphic pairs of the same order, including their adjacency matrices or vertex lists, the individual local energies, the resulting e(G) values (shown to be equal), and a brief note on how the energies were computed (via standard spectral methods). This will make the central claim verifiable and strengthen the manuscript's contribution. revision: yes
Circularity Check
No significant circularity identified
full rationale
The paper introduces no derivation chain, predictions, or fitted parameters. It adopts the external 2024 definition of local energy e(G) = n·ℰ(G) − ∑_v ℰ(G−v) from Espinal and Rada, then reports observational pairs of same-order graphs with equal e(G). The central activity is enumeration of examples satisfying an unambiguous equality condition; no step reduces to self-definition, self-citation, or renaming of inputs. The work is self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Graph energy is the sum of the absolute values of the eigenvalues of the adjacency matrix.
Reference graph
Works this paper leans on
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discussion (0)
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