TL;DR
This paper extends the transplantation method for isospectral graphs with mixed boundary conditions by introducing vertex-colored and edge-colored line graphs, linking graph theory with Brownian motion and semigroup representations.
Contribution
It generalizes the transplantation method using colored line graphs and semigroup representations, enabling the construction of new isospectral graphs with mixed boundary conditions.
Findings
Introduces vertex-colored and edge-colored line graphs for transplantation
Rephrases transplantation in terms of free semigroup representations
Provides a method to generate adjacency cospectral weighted directed graphs
Abstract
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.
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