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Graph theory and discrete mathematics · Andrea Paone, Marco Paone

This preprint studies the signature of the adjacency matrix of line graphs using the shifted signless Laplacian Q(G) − 2I, rooted-response methods, and reductions to the 2-core. It proves an exact pendant-forest reductio

Graph theory and discrete mathematics · Andrea Paone

Akbari, Elphick, Kumar, Pragada and Tang conjectured that every connected graph satisfies a one-unit upper bound between the positive and negative adjacency inertia indices of its line graph. The manuscript presents a co

Graph theory and discrete mathematics · Andrea Paone

The work presents a connected simple graph whose line graph has adjacency inertia (9, 0, 7), giving an exact counterexample to Conjecture 4.12. The public record includes an adjacency matrix, exact-arithmetic certificate

Cybersecurity and AI governance · Andrea Paone

The paper assesses whether the European Union's 2026 Action Plan on Cybersecurity and Artificial Intelligence converts an extensive regulatory framework into operational capabilities against AI-enabled cyber threats, wit

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

1 Introduction

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

2 Definitions and notation

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

3 Attachment formulas for pendant trees

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

4 Counterexamples to 2-core monotonicity

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

5 Bounds at fixed cyclomatic number

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

6 Pendant attachments on extremal cores

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

7 Computational verification

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

8 Relationship to previous work

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

9 Limitations and open problems

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

Appendix A Additional proof details

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

Appendix B Exact certificates for the principal examples

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

Appendix C Computational details and package scope

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

Author contributions

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

Funding

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

Disclosure statement

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

Use of generative AI and AI-assisted tools

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

Data availability statement

Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

Licence

Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

1 Introduction

Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

2 Preliminaries

Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

3 The seed counterexample

Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

4 Rooted-module attachment

Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

5 A signature amplifier

Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

6 Unbounded signature on a restricted graph class

Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

7 Four-subdivision transfer and prior period-four reduction

Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

8 Fixed-signature families

Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

9 Exact classification of three-cycle chains

Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

10 Reproducibility

Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

11 Discussion

Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

License

Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

Author information

Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

Declaration of generative AI and AI-assisted technologies in the manuscript preparation process

A Counterexample to a Line-Graph Inertia Conjecture

1 Introduction

A Counterexample to a Line-Graph Inertia Conjecture

2 The graph

A Counterexample to a Line-Graph Inertia Conjecture

3 Exact inertia certificate

A Counterexample to a Line-Graph Inertia Conjecture

4 Reproducibility and scope

A Counterexample to a Line-Graph Inertia Conjecture

Appendix A Adjacency matrix of the line graph

Andrea Paone

Marco Paone

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