Triangle graph connectivity as a necessary condition for spectral lower bounds: obstructions, a strong empirical conjecture, and the limits of local triangle support

We investigate the structural conditions under which local triangle support ina graph G can yield a meaningful lower bound on its algebraic connectivity λ2(G).Three failure modes are identified: insufficient edge-level triangle support (captured byτ(G) = minetri(e)), non-uniform distribution of triangle load across vertices (capturedby the participation ratio PRnorm), and disconnectivity of the triangle graph T(G), whosevertices are the edges of G with two edges adjacent if and only if they share a triangle.We show empirically that the first two conditions are individually necessary but jointlyinsufficient, and that disconnectivity of T(G) is the universal failure mode: every naturaladversarial construction that drives λ2(G) → 0 while preserving local triangle structuresystematically disconnects T(G) first. This motivates the following strong empiricalconjecture: if T(G) is connected, then λ2(G) ≥ λ2(T(G)). We verify this inequality forover 50000 random graphs and 80 targeted adversarial constructions without finding acounterexample; Kn saturates the bound exactly for all n. We further investigate threeproof routes, show that two fail structurally (the line graph route breaks outside theregular case; the signless Laplacian route fails on wheels), and identify the remainingvariational formulation as the natural open problem: is RG(Bh) ≥ λ2(T(G)) for allh ⊥1E? Together with the companion papers establishing τ(G) as a local-to-globalbridge [8] and λ2(T(G)) ≤ λ2(G) for regular graphs [9], this work completes a three-levelstructural decomposition of triangle-based spectral connectivity.