Bounds for the energy of a complex unit gain graph

Abstract

A $\mathbb{T}$-gain graph, $\Phi =(G,\phi )$, is a graph in which the function $\phi$ 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 $2\tau (G)-2c(G)\le \mathcal{E}(\Phi )\le 2\tau (G)\sqrt{\Delta (G)}$, 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.

31 pages, 4 figures