What question did this study set out to answer?

This research aims to determine how adding edges to single-root directed graphs (digraphs) affects the second smallest eigenvalue of the Laplacian matrix.

May 7, 2026Chaos An Interdisciplinary Journal of Nonlinear Science

Two sufficient conditions for altering algebraic connectivity via edge addition in single-root digraphs

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This research aims to determine how adding edges to single-root directed graphs (digraphs) affects the second smallest eigenvalue of the Laplacian matrix.
Established a spectral framework using block lower-triangular decomposition to analyze strongly connected components.
Utilized matrix perturbation theory and adjugate-matrix identities to derive conditions for changes in eigenvalues.
Conducted numerical simulations on Erdős-Rényi and scale-free networks to validate theoretical findings.
Identified two conditions for edge addition impacting λ2: one increases or maintains, while the other causes a strict decrease.
When the algebraic multiplicity of λ2 is greater than one, λ2 can remain unchanged under certain edge conditions.
Applications to Kuramoto oscillator networks confirmed that changes predicted to decrease λ2 result in slower synchronization.

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Implication

Randomized trial shows that adding edges affects algebraic connectivity in digraphs, suggesting significant implications for network dynamics.

Cite This Study

Zou et al. (2026) studied this question.

synapsesocial.com/papers/69fc2c718b49bacb8b347fd6 https://doi.org/10.1063/5.0319682

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Implication

Randomized trial shows that adding edges affects algebraic connectivity in digraphs, suggesting significant implications for network dynamics.

Two sufficient conditions for altering algebraic connectivity via edge addition in single-root digraphs

Key Points

Discussion

Implication

Cite This Study

Two sufficient conditions for altering algebraic connectivity via edge addition in single-root digraphs

Key Points

Discussion

Implication

Cite This Study

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