On -gain graphs with few positive eigenvalues.

Let $ \mathbb {T}_4=\{1,-1,\mathbf {i},-\mathbf {i}\} $ T 4 = { 1 , − 1 , i , − i } be the group of fourth roots of unit. A $ \mathbb {T}_4 $ T 4 -gain graph is a graph where each orientation of an edge is given a complex unit in $ \mathbb {T}_4 $ T 4 , which is the inverse of the complex unit assigned to the opposite orientation. In this paper, we characterize the structure of the $ \mathbb {T}_4 $ T 4 -gain graphs with exactly one positive eigenvalue and determine the $ \mathbb {T}_4 $ T 4 -gain graphs with cut vertices having exactly two positive eigenvalues. Our results extend some parallel ones about mixed graphs and signed graphs. [ABSTRACT FROM AUTHOR]