Eigenvalues of complex unit gain graphs and gain regularity

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