Eigenvalues of complex unit gain graphs and gain regularity

A complex unit gain graph (or T -gain graph) Γ = (G, γ) is a gain graph with gains in T , the multiplicative group of complex units. The T -outgain in Γ of a vertex v ∈ G is the sum of the gains of all the arcs originating in v. A T -gain graph is said to be an a-T -regular graph if the T -outgain of each of its vertices is equal to a. In this article, it is proved that a-T -regular graphs exist for every a ∈ R. This, in particular, means that every real number can be a T -gain graph eigenvalue. Moreover, denoted by Ω(a) the class of connected T-gain graphs whose largest eigenvalue is the real number a, it is shown that Ω(a) is nonempty if and only if a belongs to {0} ∪ [1, +∞). In order to achieve these results, non-complete extended p-sums and suitably defined joins of T -gain graphs are considered.