Let ${\mathbb{T}}_4=\{\pm 1,\pm i\}$ be the subgroup of fourth roots of unity inside $\mathbb{T}$ , the multiplicative group of complex units. For a ${\mathbb{T}}_4$ -gain graph $\mathsf{\Phi }=(\mathsf{\Gamma },{\mathbb{T}}_4,\phi )$ , we introduce gain functions on its line graph $\mathcal{L}(\mathsf{\Gamma })$ and on its subdivision graph $\mathcal{S}(\mathsf{\Gamma })$ . The corresponding gain graphs $\mathcal{L}(\mathsf{\Phi })$ and $\mathcal{S}(\mathsf{\Phi })$ 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 $\mathsf{\Phi }$ , and the adjacency characteristic polynomials of $\mathcal{L}(\mathsf{\Phi })$ and $\mathcal{S}(\mathsf{\Phi })$ . A suitably defined incidence matrix for ${\mathbb{T}}_4$ -gain graphs plays an important role in this context.

We prove that for two graph classes, threshold graphs and chain graphs, there exist intervals in which their distance matrix does not have any eigenvalues. We also compute the determinant of the distance matrix of both threshold and chain graphs in terms of their generating binary sequence. In this way we positively address a research problem posed in [D.P. Jacobs, V. Trevisan, F. C. Tura, Distance eigenvalue location in thresholds graphs, Proceedings of DGA, Manaus; 2013. p. 1–4].

A graph is called a chain graph if it is bipartite and the neighbourhoods of the vertices in each colour class form a chain with respect to inclusion. In this paper, we study the relation between the degree sequences and the Laplacian spectra of chain graphs. We provide explicit formulae for the Laplacian characteristic polynomial of a chain graph, and certain properties of the Laplacian spectrum that can be deduced from its degree sequence.