Bounds for the energy of a complex unit gain graph

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.