Bounds for the energy of a complex unit gain graph

TL;DR

This paper establishes bounds for the energy of complex unit gain graphs based on graph invariants like vertex cover number, odd cycles, and maximum degree, and characterizes cases of equality.

Contribution

It introduces bounds for the energy of $ ext{T}$-gain graphs and characterizes when these bounds are tight, solving a general open problem.

Findings

Bounds for energy in terms of vertex cover, odd cycles, and maximum degree.

Characterizations of graphs where bounds are tight.

Bounds in terms of spectral radius.

Abstract

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.