Triangle redundancy constrains algebraic connectivity: a spectral identity and its implications for neural circuit dynamics

The Topostability framework classifies each edge of a network by its localtriangle count tri(e) = (A2)ij , producing a Fragility Index (FI) that measuresstructural redundancy at mesoscopic scale. Prior work established that FIdistinguishes neural circuit classes and responds to targeted perturbationssuch as deep brain stimulation. Here we ask whether FI carries spectralinformation — specifically whether it constrains the algebraic connectivityλ2(L), which governs global dynamic resilience.We prove two exact results. Theorem 1: for any connected graph,tr(L · A2) = Pi d2i − 6T, a tight identity linking the Laplacian, degreesequence, and triangle count. Theorem 2: for d-regular graphs,λ2(L) ≤ (nd2 − 6T) / (d(n − d)), an upper bound that is sharp for the completegraph Kn and decreases as triangulation increases. We show via adumbbell counter-example that no analogous lower bound in terms of totaltriangle count T exists: λ2 and T are decoupled when triangles clusterin dense sub-structures separated by a bottleneck edge. The open problemof a lower bound conditioned on the minimum per-edge triangle countτ (G) = mine tri(e) is formulated precisely and connected to the Cheegerinequality.Complementing the analytical results, a large-scale simulation across3,000 random connected graphs (n ∈ {5, . . . , 15}, variable density) confirmsthat FI predicts λ2 beyond density constraints (Spearman partial ρ = −0.077,p < 0.001, robust across four density bins). A diffusion model establishes the∗Corresponding author dynamical chain FI → λ2 → perturbation return time. Application to theseven Tier 1 neural circuits from Papers 8–10 positions each circuit withinthe spectral interval predicted by Theorem 2, with selection circuits (basalganglia, amygdala) occupying the high-FI / unconstrained-λ2 region and integrationcircuits (prefrontal-limbic, hippocampus) occupying the low-FI /constrained-λ2 region.These results establish a rigorous bridge between a local topological measureand a global spectral quantity, contributing to spectral graph theoryindependently of the neuroscientific application.Keywords: algebraic connectivity, triangle redundancy, spectral graphtheory, Fragility Index, Laplacian eigenvalues, neural circuit dynamics,Topostability