Line and subdivision graphs determined by T4-gain graphs

Let T4 = (±1, ±i) be the subgroup of fourth roots of unity inside T, the multiplicative group of complex units. For a T4-gain graph Φ = (Γ,T4, ϕ), we introduce gain functions on its line graph L(Γ) and on its subdivision graph S(Γ). The corresponding gain graphs L(Φ) and S(Φ) are defined up to switching equivalence and generalize the analogous constructions for signed graphs. We discuss some spectral properties of these graphs and in particular we establish the relationship between the Laplacian characteristic polynomial of a gain graph Φ, and the adjacency characteristic polynomials of L(Φ) and S(Φ). A suitably defined incidence matrix for T4-gain graphs plays an important role in this contex

Line graphs of complex unit gain graphs with least eigenvalue -2

Let T be the multiplicative group of complex units, and let L(φ) denote a line graph of a T-gain graph φ. Similarly to what happens in the context of signed graphs, the real number min Spec(A(L(φ)), that is, the smallest eigenvalue of the adjacency matrix of L(φ), is not less than -2. The structural conditions on φ ensuring that min Spec(A(L(φ)) = -2 are identified. When such conditions are fulfilled, bases of the -2-eigenspace are constructed with the aid of the star complement technique

On eigenspaces of some compound complex unit gain graphs

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

Spectral Fundamentals and Characterizations of Signed Directed Graphs

The spectral properties of signed directed graphs, which may be naturally obtained by assigning a sign to each edge of a directed graph, have received substantially less attention than those of their undirected and/or unsigned counterparts. To represent such signed directed graphs, we use a striking equivalence to $\mathbb{T}_6$-gain graphs to formulate a Hermitian adjacency matrix, whose entries are the unit Eisenstein integers $\exp(k\pi i/3),$ $k\in \mathbb{Z}_6.$ Many well-known results, such as (gain) switching and eigenvalue interlacing, naturally carry over to this paradigm. We show that non-empty signed directed graphs whose spectra occur uniquely, up to isomorphism, do not exist, but we provide several infinite families whose spectra occur uniquely up to switching equivalence. Intermediate results include a classification of all signed digraphs with rank $2,3$, and a deep discussion of signed digraphs with extremely few (1 or 2) non-negative (eq. non-positive) eigenvalues