Nonormality, optimality and synchronization

The paper defines ε-Laplacian pseudospectra Φε(L), the ε-pseudospectral abscissa ζε(L), and Laplacian pseudospectral resilience LPR = −ζλ2(L)/λ2, and explicitly states that for normal graph Laplacians ζλ2 = 0 (hence LPR = 0), while non-normal Laplacians may have ζλ2 < 0, supporting the claim that LPR distinguishes stability within isospectral ‘optimal’ networks; this is backed by numerical evidence across a toy family (chain→star) with identical nonzero eigenvalues . The model’s solution supplies a largely correct proof sketch for the normal case and an explicit 3×3 counterexample demonstrating that two isospectral Laplacians (one normal, one with a Jordan block) yield different LPR. Minor notational slips in the model (confusing I_{n−1} with the projector Π) are easily repaired. Overall, both are correct, with the paper offering definitions and empirical evidence and the model giving a concise proof and example.