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.