Eigenvalues of complex unit gain graphs and gain regularity

Abstract

Abstract 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 ∈ G v\in G is the sum of the gains of all the arcs originating in v v . A T {\mathbb{T}} -gain graph is said to be an a a - T {\mathbb{T}} -regular graph if the T {\mathbb{T}} -outgain of each of its vertices is equal to a a . In this article, it is proved that a a - T {\mathbb{T}} -regular graphs exist for every a ∈ R a\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 a a , it is shown that Ω ( a ) \Omega \left(a) is nonempty if and only if a a belongs to { 0 } ∪ [ 1 , + ∞ ) \left\{0\right\}\cup \left[1,+\infty ) . In order to achieve these results, non-complete extended p p -sums and suitably defined joins of T {\mathbb{T}} -gain graphs are considered.