math.CO — Combinatorics
Kernel Dimension of the Z/2 Connection Laplacian on a Finite Simple Graph (A Formalisation in Lean 4 / Mathlib v4.11.0)
We present a Lean 4/Mathlib v4.11.0 formalization of the kernel-dimension theory of the connection Laplacian on finite simple graphs equipped with designated wrap edges. Each edge carries either the identity or the coordinate-swap matrix. A Hadamard–Kronecker change of basis decomposes the resulting rank-two connection Laplacian into the ordinary and signed graph Laplacians. An analogous splitting holds for the scalar Laplacian of the associated two-sheeted cover. A connected component is balanced precisely when its covering preimage has two connected components, equivalently when its sign holonomy is trivial along every closed walk. Consequently, the nullity of the signed Laplacian equals the number of balanced connected components, while the nullity of the full connection Laplacian equals the number of connected components plus the number of balanced components. For the flat connection, the nullity is twice the number of connected components. We also formalize exact characteristic polynomials and eigenvalue multisets, including multiplicities, for explicit ordinary and one-wrap signed cycle matrices for . Transfer to general connection-graph encodings remains conditional on specified matrix-reindexing identities. The contribution is a machine-checked formalization, exposition, and documented verification of results rooted in classical signed-graph spectral theory, rather than a claim of priority for those identities. The accompanying repository preserves proof artifacts, computational cross-checks, historical verification evidence, and documented corrections.