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 (capturedby 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 atriangle. 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 followingstrong empirical conjecture: if T(G) is connected, then λ2(G) ≥ λ2(T(G)). We verify this inequality for over 50 000 random graphs and 80 targeted adversarial constructionswithout finding a counterexample; Kn saturates the bound exactly for all n. We further investigate three proof routes, show that two fail structurally (the line graphroute breaks outside the regular case; the signless Laplacian route fails on wheels), and identify the remaining variational formulation as the natural open problem: isRG(Bh) ≥ λ2(T(G)) for all h ⊥ 1E? Together with the companion papers establishing τ(G) as a local-to-global bridge 9 and λ2(T(G)) ≤ λ2(G) for regular graphs 10 (nowformally verified in Lean 4 7, including the Cheeger hard direction), this work completes a three-level structural decomposition of triangle-based spectral connectivity. In this updated version (v3), we report the refutation of the raw conjecture τ(G) ≤ λ2(G) for irregular graphs, propose the degree-corrected replacement τ(G)/(∆ − 1) ≤ λ2(G),and show that the two lower bounds on λ2(G) — the local τ/(∆−1) and the global λ2(T(G)) — are independent: neither implies the other, yet both hold empirically onover 72000 connected graphs.