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 a
graph G and whose edges connect pairs that share a common triangle. The second Laplacian
eigenvalue 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 is
elementary: 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 standard
spectral bound λ2 ≤ d+1. No symmetry assumption, no Cheeger inequality, and no positivesemidefinite
dominance argument is required.
The bound is sharp: Kn is the unique saturating family, with λ2(T(Kn)) = λ2(Kn) for all
n ≥ 3. We characterise further exact classes: strongly regular graphs satisfy Ltrian = λ · LG
exactly, and triangle-free graphs give λ2(T(G)) = 0. For vertex-transitive graphs we exhibit
additional algebraic structure: Ltrian and LG commute, and the inequality can be proved
independently via a commutation argument on the automorphism algebra.
Compared with the combinatorial route of a companion paper [7] (the τ (G)-based lower
bound via Cheeger’s inequality), the present spectral bridge is direct, tight at Kn, and
bypasses the two min/sum mismatches responsible for a ×500–1300 empirical slack.