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 $G$.