Some New Results on Energy of Graphs with Self Loops

Kalpesh M. Popat; Kunal R. Shingala

arxiv: 2604.18651 · v1 · submitted 2026-04-20 · 🧮 math.CO · math.SP

Some New Results on Energy of Graphs with Self Loops

Kalpesh M. Popat , Kunal R. Shingala This is my paper

Pith reviewed 2026-05-10 04:44 UTC · model grok-4.3

classification 🧮 math.CO math.SP

keywords graph energyself-loopseigenvaluesgraph spectraenergy invariancelooped graphs

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The pith

There are graphs for which the energy remains unchanged even after attaching self-loops to some but not all vertices.

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

This paper introduces a definition for the energy of a graph with self-loops by shifting each eigenvalue by the average number of loops per vertex. It establishes that the energy matches the loop-free case when loops are attached to every vertex or to none. The authors then answer whether equality can hold for intermediate numbers of loops by exhibiting a family of graphs where this is true. A reader would care because this shows the new energy can coincide with the classical one in non-trivial cases, potentially making it a consistent extension.

Core claim

The central claim is that there exists a family of graphs G such that for σ between 1 and n-1, the energy E(G_σ) defined using the shifted eigenvalues equals the energy of the original graph G.

What carries the argument

The shifted energy sum |λ_i - σ/n| over the eigenvalues of the graph with σ self-loops, which reduces to the standard energy sum |λ_i| when σ is zero or n.

If this is right

If σ equals zero or n, the equality E(G) = E(G_σ) holds for any graph G.
The authors provide an explicit family of graphs that satisfy E(G) = E(G_σ) for 0 < σ < n.
This equality means the total absolute deviation from the shift is the same as without the shift for these graphs.
The family serves as examples where partial self-loop addition preserves energy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These graphs likely have eigenvalue sets whose absolute deviations are unaffected by the uniform shift.
This invariance might extend to other spectral measures or graph modifications.
One could test if random graphs or other common families exhibit similar behavior.

Load-bearing premise

The definition of the modified energy using the absolute deviation from σ/n captures a useful notion of energy for graphs with self-loops.

What would settle it

Computing the eigenvalues of the graphs in the family and checking if sum |λ_i| equals sum |λ_i - σ/n| for the relevant σ.

read the original abstract

The graph $G_\sigma$ is obtained from graph $G$ by attaching self loops on $\sigma$ vertices. The energy $ E(G_\sigma)$ of the graph $G_\sigma$ with order $n$ and eigenvalues $\lambda_1,\lambda_2,\dots,\lambda_n$ is defined as $ E(G_\sigma)= \displaystyle \sum_{i=1}^n\left|\lambda_i-\dfrac{\sigma}{n}\right| $. It has been proved that if $\sigma=0\; or\; n$ then $ E(G)=E(G_\sigma) $. The obvious question arise: Are there any graph such that $E(G)=E(G_\sigma)$ and 0$<\sigma<n$? We have found an affirmative answer of this question and contributed a graph family which satisfies this property.

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

The paper constructs a family of graphs where adding self-loops to a proper subset of vertices leaves the energy unchanged under their centered definition.

read the letter

The main takeaway is that the authors answer their own question with an explicit family of graphs G where E(G) equals the centered E(G_σ) for some 0 < σ < n. They start from the known boundary cases and show the equality can hold in between as well. The centering step subtracts σ/n from each eigenvalue, which matches the trace of the adjacency matrix and keeps the adjustment consistent with the mean shift from the loops. If the construction is fully spelled out and the eigenvalues check, this settles the existence claim they pose in the abstract. That is the concrete piece they deliver. The work is narrow but direct: it gives examples rather than a broad theory. The soft spot is the lack of context around the definition itself. Subtracting the average eigenvalue is a clean move on paper, but the manuscript does not explain why this particular centering is the right extension for graphs with loops or how it compares to other energy variants in the literature. Without that discussion or a check against known equienergetic constructions, the family reads as a modest tweak rather than a deeper advance. The paper is aimed at people who already work on spectral graph theory and graph energies. A reader in that niche can pull the family for their own calculations or counterexamples. It is focused enough to deserve a referee who can verify the eigenvalue details and ask for the missing comparisons. I would send it to peer review. The claim is limited but verifiable, and a referee can sort out whether the construction holds and whether the motivation needs strengthening.

Referee Report

2 major / 2 minor

Summary. The manuscript defines G_σ as the graph obtained from G by adding self-loops to σ vertices. It defines the energy E(G_σ) = ∑ |λ_i − σ/n| where λ_i are the eigenvalues of the adjacency matrix of G_σ. It proves that E(G) = E(G_σ) when σ = 0 or σ = n, and claims to construct an infinite family of graphs G for which the equality holds for some 0 < σ < n.

Significance. If the claimed family is explicitly constructed and the eigenvalue compensation is verified, the result would show that the centering shift by the mean eigenvalue can leave the L1-norm of the spectrum unchanged for non-trivial looped graphs. This would be a modest but concrete addition to the literature on graph energies and spectral invariants under local modifications such as self-loops. The boundary-case proofs already supplied constitute a clear foundation.

major comments (2)

[Abstract] Abstract: the central claim asserts an affirmative answer together with an explicit graph family, yet the manuscript supplies neither the family description, the adjacency matrix of any member, nor the eigenvalue calculation demonstrating that the deviations from σ/n exactly compensate to recover the unshifted energy sum. This verification is load-bearing for the main result.
[Definition of E(G_σ)] The definition E(G_σ) = ∑ |λ_i − σ/n| is introduced without deriving or exhibiting the spectrum of A(G_σ) relative to A(G), leaving open whether any non-boundary equality can hold beyond the trace identity already noted.

minor comments (2)

[Abstract] Grammatical error: 'The obvious question arise' should read 'arises'.
[Abstract] Notation inconsistency: '0$<σ<n$' mixes plain text and inline LaTeX; use consistent mathematical formatting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of the central claim. We agree that the submitted version did not sufficiently detail the graph family, its adjacency matrices, or the eigenvalue compensation, and we will revise the manuscript to include these elements.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim asserts an affirmative answer together with an explicit graph family, yet the manuscript supplies neither the family description, the adjacency matrix of any member, nor the eigenvalue calculation demonstrating that the deviations from σ/n exactly compensate to recover the unshifted energy sum. This verification is load-bearing for the main result.

Authors: We acknowledge that the abstract announces an explicit infinite family but the submitted manuscript did not provide a concrete description, an example adjacency matrix, or the explicit eigenvalue list showing exact compensation. In the revised version we will add a dedicated section that (i) defines the family via a recursive or parametric construction, (ii) exhibits the adjacency matrix of the smallest member, and (iii) computes its spectrum of A(G_σ) relative to A(G) to verify that the absolute deviations from σ/n sum to the same value as the unshifted energy. revision: yes
Referee: [Definition of E(G_σ)] The definition E(G_σ) = ∑ |λ_i − σ/n| is introduced without deriving or exhibiting the spectrum of A(G_σ) relative to A(G), leaving open whether any non-boundary equality can hold beyond the trace identity already noted.

Authors: The definition follows directly from the observation that the trace of A(G_σ) equals σ (each self-loop contributes 1), hence the average eigenvalue is σ/n; the energy is then the L1 deviation from this mean. The boundary proofs for σ = 0 and σ = n are already given. For the non-boundary case we will supply, in the revision, the explicit spectrum of the constructed family together with the algebraic verification that the positive and negative deviations cancel in the L1 sense, thereby establishing that equality holds for 0 < σ < n. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces a definition of centered energy E(G_σ) = ∑ |λ_i - σ/n| for graphs with σ self-loops, proves the boundary cases σ=0 and σ=n reduce to ordinary energy by direct substitution into the definition, and then constructs an explicit infinite family of graphs where the equality E(G)=E(G_σ) holds for intermediate 0<σ<n. No parameters are fitted to data, no results are renamed as predictions, and the central existence claim rests on an explicit construction rather than any self-referential equation or load-bearing self-citation. The derivation chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition of graph energy via adjacency eigenvalues and the algebraic effect of adding self-loops (which adds 1 to selected diagonal entries). No free parameters, new entities, or non-standard axioms are introduced in the abstract.

axioms (2)

standard math The energy of a graph is the sum of absolute values of the eigenvalues of its adjacency matrix.
Invoked implicitly when comparing E(G) and E(G_σ).
domain assumption Adding a self-loop to a vertex increases the corresponding diagonal entry of the adjacency matrix by 1.
Used to define G_σ from G.

pith-pipeline@v0.9.0 · 5441 in / 1325 out tokens · 47478 ms · 2026-05-10T04:44:23.329733+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Adiga, R

C. Adiga, R. Balakrishnan, W. So, The Skew Energy of a Digraph, Linear Algebra Appl.432(2010) 1825–1835

work page 2010
[2]

Cvetkovi`c, P

D. Cvetkovi`c, P. Rowlison, S. Simi`c,An Introduction to the Theory of Graph Spectra, Cambridge university press, 2010

work page 2010
[3]

Indulal, A

G. Indulal, A. Vijaykumar, A Note on Energy of Some Graphs, MATCH Commun. Math.Comput. Chem.59(2008) 269 - 274

work page 2008
[4]

Gutman, The energy of a graph,Ber

I. Gutman, The energy of a graph,Ber. Math. Statist. Sekt. Forsch- ungszentram Graz.103(1978) 1 - 22

work page 1978
[5]

Gutman, I

I. Gutman, I. Redˇzepoviˇc, B. Furtula, A. M. Sahal, Energy of graphs with self-loops,MATCH Commun. Math. Comput. Chem.87(2022) 645 - 652

work page 2022
[6]

Jovanovi`c, E

I. Jovanovi`c, E. Zogi`c, E. Glogi`c, On the Conjecture Related to the Energy of Graphs with Self–Loops,MATCH Commun. Math. Com- put. Chem.89(2023) 479 - 488

work page 2023
[7]

Gutman, D

I. Gutman, D. kiani, M. Mirazakhah,B. Zhou, On Incidence Energy of Graph,Linear Algebra Appl.431(2009), 1223–1233. 9

work page 2009
[8]

Balakrishnan, The Energy of graph,Linear Algebra Appl

R. Balakrishnan, The Energy of graph,Linear Algebra Appl. 387(2004), 287–295

work page 2004
[9]

Balakrishnan, K

R. Balakrishnan, K. Ranganathan,A Textbook of Graph Theory, Springer, New York, 2000

work page 2000
[10]

R. B. Bapat, S. Pati, Energy of a graph is never an odd integer,Bull. Kerala Math. Assoc.1(2004) 129–132

work page 2004
[11]

Lang,Algebra, Springer, New York, 2002

S. Lang,Algebra, Springer, New York, 2002

work page 2002
[12]

S. B. Bozkurt, A. D. Gungor, I Gutman, Note on Distance Energy of Graphs,SIAM J. Discrete Math.64(2010) 129–134

work page 2010
[13]

X. Li, Y. Shi, I. Gutman,Graph Energy, Springer, New York, 2012

work page 2012

[1] [1]

Adiga, R

C. Adiga, R. Balakrishnan, W. So, The Skew Energy of a Digraph, Linear Algebra Appl.432(2010) 1825–1835

work page 2010

[2] [2]

Cvetkovi`c, P

D. Cvetkovi`c, P. Rowlison, S. Simi`c,An Introduction to the Theory of Graph Spectra, Cambridge university press, 2010

work page 2010

[3] [3]

Indulal, A

G. Indulal, A. Vijaykumar, A Note on Energy of Some Graphs, MATCH Commun. Math.Comput. Chem.59(2008) 269 - 274

work page 2008

[4] [4]

Gutman, The energy of a graph,Ber

I. Gutman, The energy of a graph,Ber. Math. Statist. Sekt. Forsch- ungszentram Graz.103(1978) 1 - 22

work page 1978

[5] [5]

Gutman, I

I. Gutman, I. Redˇzepoviˇc, B. Furtula, A. M. Sahal, Energy of graphs with self-loops,MATCH Commun. Math. Comput. Chem.87(2022) 645 - 652

work page 2022

[6] [6]

Jovanovi`c, E

I. Jovanovi`c, E. Zogi`c, E. Glogi`c, On the Conjecture Related to the Energy of Graphs with Self–Loops,MATCH Commun. Math. Com- put. Chem.89(2023) 479 - 488

work page 2023

[7] [7]

Gutman, D

I. Gutman, D. kiani, M. Mirazakhah,B. Zhou, On Incidence Energy of Graph,Linear Algebra Appl.431(2009), 1223–1233. 9

work page 2009

[8] [8]

Balakrishnan, The Energy of graph,Linear Algebra Appl

R. Balakrishnan, The Energy of graph,Linear Algebra Appl. 387(2004), 287–295

work page 2004

[9] [9]

Balakrishnan, K

R. Balakrishnan, K. Ranganathan,A Textbook of Graph Theory, Springer, New York, 2000

work page 2000

[10] [10]

R. B. Bapat, S. Pati, Energy of a graph is never an odd integer,Bull. Kerala Math. Assoc.1(2004) 129–132

work page 2004

[11] [11]

Lang,Algebra, Springer, New York, 2002

S. Lang,Algebra, Springer, New York, 2002

work page 2002

[12] [12]

S. B. Bozkurt, A. D. Gungor, I Gutman, Note on Distance Energy of Graphs,SIAM J. Discrete Math.64(2010) 129–134

work page 2010

[13] [13]

X. Li, Y. Shi, I. Gutman,Graph Energy, Springer, New York, 2012

work page 2012