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 in
a 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 by
τ(G) = minetri(e)), non-uniform distribution of triangle load across vertices (captured
by the participation ratio PRnorm), and disconnectivity of the triangle graph T(G), whose
vertices 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 jointly
insufficient, and that disconnectivity of T(G) is the universal failure mode: every natural
adversarial construction that drives λ2(G) → 0 while preserving local triangle structure
systematically disconnects T(G) first. This motivates the following strong empirical
conjecture: if T(G) is connected, then λ2(G) ≥ λ2(T(G)). We verify this inequality for
over 50000 random graphs and 80 targeted adversarial constructions without finding a
counterexample; Kn saturates the bound exactly for all n. We further investigate three
proof routes, show that two fail structurally (the line graph route breaks outside the
regular case; the signless Laplacian route fails on wheels), and identify the remaining
variational formulation as the natural open problem: is RG(Bh) ≥ λ2(T(G)) for all
h ⊥1E? Together with the companion papers establishing τ(G) as a local-to-global
bridge [8] and λ2(T(G)) ≤ λ2(G) for regular graphs [9], this work completes a three-level
structural decomposition of triangle-based spectral connectivity.