Triangle-connectivity graphs and a spectral dominance conjecture for regular graphs

We introduce the triangle-connectivity graph T(G), whose vertices are the edges of agraph G and whose edges connect pairs that share a common triangle. The second Laplacianeigenvalue λ2(T(G)) captures the global cohesion of the triangular structure of G.We prove that λ2(T(G)) ≤ λ2(G) for every connected d-regular graph. The proof iselementary: it combines a variational reduction via the test vector h({u, v}) = f(u) + f(v)(where f is the Fiedler eigenvector of G), the local bound tri(e) ≤ d − 1, and the standardspectral bound λ2 ≤ d+1. No symmetry assumption, no Cheeger inequality, and no positivesemidefinitedominance argument is required.The bound is sharp: Kn is the unique saturating family, with λ2(T(Kn)) = λ2(Kn) for alln ≥ 3. We characterise further exact classes: strongly regular graphs satisfy Ltrian = λ · LGexactly, and triangle-free graphs give λ2(T(G)) = 0. For vertex-transitive graphs we exhibitadditional algebraic structure: Ltrian and LG commute, and the inequality can be provedindependently via a commutation argument on the automorphism algebra.Compared with the combinatorial route of a companion paper [7] (the τ (G)-based lowerbound via Cheeger’s inequality), the present spectral bridge is direct, tight at Kn, andbypasses the two min/sum mismatches responsible for a ×500–1300 empirical slack.