The hydrogen identity for Laplacians

For any 1-dimensional simplicial complex G defined by a finite simple graph, the hydrogen identity |H| = L−L−1 holds, where |H| = (|d| + |d|∗)2 is the sign-less Hodge Laplacian defined by the sign-less incidence matrix |d| and where L is the connection Laplacian. Having linked the Laplacian spectral radius ρ of G with the spectral radius of the adjacency matrix its connection graph G′ allows for every k to estimate ρ ≤ rk − 1/rk, where rk = 1 + (P (k)) 1/k and P (k) = maxxP (k, x), where P (k, x) is the number of paths of length k starting at a vertex x in G′. The limit rk − 1/rk for k → ∞ is the spectral radius ρ of |H| which by Wielandt is an upper bound for the spectral radius ρ of H = (d + d∗)2, with equality if G is bipartite. We can relate so the growth rate of the random walks in the line graph GL of G with the one in the connection graph G′ of G. The hydrogen identity implies that the random walk ψ(n) = Lψ on the connection graphG′ with integer n solves the 1-dimensional Jacobi equation ∆ψ = |H|ψ with ∆u(n) = u(n+ 2)− 2u(n) + u(n− 2) and assures that every solution is represented by such a reversible path integral. The hydrogen identity also holds over any finite field F . There, the dynamics Lψ with n ∈ Z is a reversible cellular automaton with alphabet F. By taking products of simplicial complexes, such processes can be defined over any lattice Z. Since L and L−2 are isospectral, by a theorem of Kirby, L is always similar to a symplectic matrix if the graph has an even number of simplices. By the implicit function theorem, the hydrogen relation is robust in the following sense: any matrix K with the same support than |H| can still be written as K = L − L−1 with a connection Laplacian satisfying L(x, y) = L−1(x, y) = 0 if x ∩ y = ∅.