The Shrödinger operator on graphs and topology

We shall define a Schrödinger operator for a one-dimensional simplicial complex (a graph) 1 Γ without ends (that is, at least two and only finitely many edges meet at any vertex), which acts on functions of vertices T or edges R: (Lψ) T = T ′ b T :T ′ ψ T ′ (V) for L 0 = ∂∂ * , b T :T ′ = 1, b T :T ′ = −m T , Here the coefficients are real, symmetric, and non-zero only for nearest neighbours T ′ ∪ T = ∂R, R ∩ R ′ = T. The coefficients b T :T = W T and d R:R = W R are called the potential, m T is the number of edges at a vertex, and ∂ is the boundary operator.