Bounds and extremal graphs for the energy of complex unit gain graphs

A complex unit gain graph ($\mathbb{T}$-gain graph), $\Phi=(G, \varphi)$ is a graph where the gain function $\varphi$ assigns a unit complex number to each orientation of an edge of $G$ 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 $A(\Phi)$. For any connected triangle-free $\mathbb{T}$-gain graph $\Phi$ with the minimum vertex degree $\delta$, we establish a lower bound $\mathcal{E}(\Phi)\geq 2\delta$ and characterize the equality. Then, we present a relationship between the characteristic and the matching polynomial of $\Phi$. Using this, we obtain an upper bound for the energy $\mathcal{E}(\Phi)\leq 2\mu\sqrt{2\Delta_e+1}$ and characterize the classes of graphs for which the bound sharp, where $\mu$ and $\Delta_e$ are the matching number and the maximum edge degree of $\Phi$, respectively. Further, for any unicyclic graph $G$, we study the gains for which the gain energy $\mathcal{E}(\Phi)$ attains the maximum/minimum among all $\mathbb{T}$-gain graphs defined on $G$.