Construction of cospectral graphs, signed graphs and T-gain graphs via partial transpose

In the wake of Dutta and Adhikari, who in 2020 used partial transposition in order to get pairs of cospectral graphs, we investigate partial transposition for Hermitian complex matrices. This allows us to construct infinite pairs of complex unit gain graphs (or T-gain graphs) sharing either the spectrum of the adjacency matrix or even the spectrum of all the generalized adjacency matrices. This investigation also sheds new light on the classical case, producing examples that were still missing even for graphs. Partial transposition requires a block structure of the matrix: we interpreted it as if coming from a composition of T-gain digraphs. By proposing a suitable definition of rigidity specifically for T-gain digraphs, we then produce the first examples of pairs of non-isomorphic graphs, signed graphs and T-gain graphs obtained via partial transposition of matrices whose blocks form families of commuting normal matrices. In some cases, the non-isomorphic graphs detected in this way turned out to be hardly distinguishable, since they share the adjacency, the Laplacian and the signless Laplacian spectrum, together with many non-spectral graph invariants.