On eigenspaces of some compound complex unit gain graphs

Let \mathbb T be the multiplicative group of complex units, and let L(\Phi) denote the Laplacian matrix of a nonempty \mathbb{T}-gain graph \Phi=(\Gamma, \mathbb{T}, \gamma). The gain line graph \mathcal L(\Phi) and the gain subdivision graph \mathcal S(\Phi) are defined up to switching equivalence. We discuss how the eigenspaces determined by the adjacency eigenvalues of \mathcal L(\Phi) and \mathcal S(\Phi) are related with those of L(\Phi).

Keywords:

Complex unit gain graph , line graph , subdivision graph , oriented gain graph , voltage graph