On the wreath product of signed and gain graphs and its spectrum

We introduce a notion of wreath product of two gain graphs (Γ1, ψ1, G1) and (Γ2, ψ2, G2), producing a gain graph over the direct product group G2|VΓ1| × G1, whose underlying graph is the classical wreath product of graphs Γ1≀Γ2. By composition with a suitable group homomorphism, our construction produces a signed graph when the two factors are signed graphs. We prove that the wreath product is stable under switching isomorphism. By using group representations, we are able to perform spectral computations on the wreath product: in particular, we determine its largest and its smallest eigenvalue, and we give a description of the spectrum when the first factor is a complex unit complete balanced or antibalanced gain graph, and the second factor is circulant. Finally, when G1 is a group of permutations of the vertex set of the first factor, and the group G2 is abelian, we give an alternative definition producing a gain graph over the group wreath product G1≀G2, which turns out to be stable under switching equivalence of the second factor, when the first factor is balanced.