  A $\mathbb{T}$-gain graph, $\Phi = (G, \varphi)$, is a graph in which the
function $\varphi$ assigns a unit complex number to each orientation of an
edge, and its inverse is assigned to the opposite orientation. The associated
adjacency matrix $ A(\Phi) $ is defined canonically. The energy $
\mathcal{E}(\Phi) $ of a $ \mathbb{T} $-gain graph $ \Phi $ is the sum of the
absolute values of all eigenvalues of $ A(\Phi) $.
  We study the notion of energy of a vertex of a $ \mathbb{T} $-gain graph, and
establish bounds for it. For any $ \mathbb{T} $-gain graph $ \Phi$, we prove
that $2\tau(G)-2c(G) \leq \mathcal{E}(\Phi) \leq 2\tau(G)\sqrt{\Delta(G)}$,
where $ \tau(G), c(G)$ and $ \Delta(G)$ are the vertex cover number, the number
of odd cycles and the largest vertex degree of $ G $, respectively.
Furthermore, using the properties of vertex energy, we characterize the classes
of $ \mathbb{T} $-gain graphs for which $ \mathcal{E}(\Phi)=2\tau(G)-2c(G) $
holds. Also, we characterize the classes of $ \mathbb{T} $-gain graphs for
which $\mathcal{E}(\Phi)= 2\tau(G)\sqrt{\Delta(G)} $ holds. This
characterization solves a general version of an open problem. In addition, we
establish bounds for the energy in terms of the spectral radius of the
associated adjacency matrix.
