Line graphs and the transplantation method

We study isospectrality for mixed Dirichlet–Neumann boundary conditions and extend the previously derived graph-theoretic formulation of the transplantation method. Led by the theory of Brownian motion, we introduce vertex-colored and edge-colored line graphs that give rise to block diagonal transplantation matrices. In particular, we rephrase the transplantation method in terms of representations of free semigroups and provide a method for generating adjacency cospectral weighted directed graphs.