On Spectra of $\mathbb{T}$-Gain Digraphs

Abstract

A \(\mathbb{T}\)-gain digraph is a directed graph with complex unit gains on its arcs, allowing for no restrictions on oppositely directed arcs. This framework unifies various graph types, such as signed graphs, mixed graphs, complex unit gain graphs, digraphs, and signed digraphs. The gain adjacency, Laplacian, and signless Laplacian matrices are generally non-Hermitian with complex spectra. We develop the spectral theory of these matrices, extending classical results from signed digraphs and complex unit-gain graphs. For the gain adjacency matrix, we establish determinant and characteristic polynomial formulas, characterize cycle balance through switching equivalence and cospectrality with the underlying digraph. We further bound the spectral radius in terms of the underlying digraph and its maximum out-degree, with equality characterized by $μ$-balance. As a consequence, we determine the spectra of $\mathbb{T}$-gain unicyclic digraphs. Moreover, we characterize cycle balance and antibalance for the Laplacian and signless Laplacian matrices, respectively, through the presence of a zero eigenvalue.