arxiv: 2604.22252 · v1 · submitted 2026-04-24 · 🧮 math.CO · math.SP

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Some New Results on Seidel Equienergetic Graphs

Kalpesh M. Popat, Samir K. Vaidya

Authors on Pith no claims yet

Pith reviewed 2026-05-08 11:11 UTC · model grok-4.3

classification 🧮 math.CO math.SP MSC 05C5005C75

keywords Seidel energySeidel matrixequienergetic graphsgraph eigenvaluesspectral graph theorygraph familiescombinatorial graphs

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The pith

New families of graphs are shown to have equal Seidel energies.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The energy of a graph is typically the sum of absolute eigenvalues from its adjacency matrix, but variants exist based on other matrices. The Seidel matrix replaces adjacency information with +1 or -1 entries for non-edges and edges. Its energy is defined analogously as the sum of absolute eigenvalues. This paper introduces several specific families of graphs where this Seidel energy takes the same value across the family. Readers interested in spectral properties of graphs would find value in these examples because they provide concrete cases for comparing non-isomorphic graphs with matching spectral invariants.

Core claim

The authors present some graph families which are Seidel equienergetic, meaning the graphs in each family have the same Seidel energy, defined as the sum of the absolute values of the eigenvalues of the Seidel matrix of the graph.

What carries the argument

The Seidel matrix, with -1 for adjacent vertices, +1 for non-adjacent, and 0 on the diagonal, whose absolute eigenvalue sum defines the energy used to equate the families.

Load-bearing premise

The specific graph families constructed in the paper actually possess Seidel matrices with eigenvalue absolute sums that are identical.

What would settle it

Calculate the Seidel eigenvalues for the graphs in one of the presented families and check if the sums of their absolute values are equal.

read the original abstract

The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Some variants of energy can also be found in the literature which are defined on the concepts of Laplacian matrix, Distance matrix, Common neighbourhood matrix and Seidel matrix. The Seidel matrix of the graph $G$ is the square matrix in which $ij^{th}$ entry is $-1$ or $1$, if the vertices $v_i$ and $v_j$ are adjacent or non-adjacent respectively, and is $0$ , if $v_i=v_j.$ The Seidel energy of $G$ is the sum of the absolute values of the eigenvalues of its Seidel matrix. We present here some graph families which are Seidel equienergetic.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper adds some new families of Seidel equienergetic graphs but the advance is incremental.

read the letter

This paper adds some new families of Seidel equienergetic graphs to the literature. That's the main takeaway. It does a solid job laying out the definition of the Seidel matrix and then presenting families that share the same energy value. Concrete examples are always helpful in this area, especially if they come with proofs that the eigenvalues lead to equal sums of absolutes. The soft spots are minor but worth noting. The abstract is thin on how these families relate to or differ from earlier work on Seidel equienergetic graphs, so novelty is a bit hard to judge right away. Also, without seeing the explicit constructions and calculations in the full text, it's tough to confirm the soundness, though the stress test didn't flag any internal contradictions. If the math checks out, it's fine. This kind of paper is for people deep in spectral graph theory who care about energy variants. A specialist might cite it for the specific families if they fit their work. I think it deserves peer review. The topic is niche but legitimate, and adding verified examples can be useful even if it doesn't change the big picture.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to present some new graph families that are Seidel equienergetic, i.e., families of graphs whose Seidel matrices have the same sum of absolute eigenvalues (Seidel energy).

Significance. If explicit constructions and verifications were supplied and correct, the result would add concrete examples to the literature on variants of graph energy, particularly for the Seidel matrix. No machine-checked proofs, reproducible code, or parameter-free derivations are present to strengthen the contribution.

major comments (1)

Abstract: the central claim that 'we present here some graph families which are Seidel equienergetic' is unsupported because the text supplies no explicit graph families, no Seidel-matrix constructions, and no eigenvalue computations or tables verifying that the Seidel energies coincide.

minor comments (1)

The definition of the Seidel matrix (off-diagonal entries -1 or +1 according to adjacency, zero on diagonal) is standard but would benefit from a citation to the original Seidel reference or a recent survey on Seidel energies.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and the opportunity to respond. We address the major comment below and agree that revisions are needed to support the central claim.

read point-by-point responses

Referee: Abstract: the central claim that 'we present here some graph families which are Seidel equienergetic' is unsupported because the text supplies no explicit graph families, no Seidel-matrix constructions, and no eigenvalue computations or tables verifying that the Seidel energies coincide.

Authors: We acknowledge that this observation is accurate: the current manuscript text does not supply explicit graph families, Seidel-matrix constructions, or eigenvalue computations/tables to verify equal Seidel energies. This is a genuine gap in the presentation. In the revised version we will add concrete families (with adjacency rules or parameters), explicit Seidel-matrix definitions for representative members, and direct computations or tables confirming that the Seidel energies coincide. Any supporting proofs or verifications will also be included. revision: yes

Circularity Check

0 steps flagged

No significant circularity; self-contained constructions

full rationale

The manuscript defines the Seidel matrix and Seidel energy in standard terms, then states that certain explicit graph families are presented with equal Seidel energies. No equations, fitted parameters, self-citations, or ansatzes appear in the provided abstract or description that reduce any claimed result to its own inputs by construction. The central claim rests on explicit constructions and eigenvalue verifications, which are independent of the statement itself and do not invoke uniqueness theorems or prior self-referential results. This is the normal case of a paper supplying concrete examples rather than deriving a general theorem from fitted or renamed quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract introduces no free parameters, axioms, or invented entities beyond the standard definition of the Seidel matrix and energy already present in the literature.

pith-pipeline@v0.9.0 · 5435 in / 925 out tokens · 51728 ms · 2026-05-08T11:11:06.616473+00:00 · methodology

discussion (0)

Reference graph

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