Bounds for the energy of a complex unit gain graph

A $\mathbb{T}$-gain graph, $Φ= (G, φ)$, is a graph in which the function $φ$ assigns a unit complex number to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix $A(Φ)$ is defined canonically. The energy $\mathcal{E}(Φ)$ of a $\mathbb{T}$-gain graph $Φ$ is the sum of the absolute values of all eigenvalues of $A(Φ)$. 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 $Φ$, we prove that $2τ(G)-2c(G) \leq \mathcal{E}(Φ) \leq 2τ(G)\sqrt{Δ(G)}$, where $τ(G), c(G)$ and $Δ(G)$ 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 $\mathcal{E}(Φ)=2τ(G)-2c(G)$ holds. Also, we characterize the classes of $\mathbb{T}$-gain graphs for which $\mathcal{E}(Φ)= 2τ(G)\sqrt{Δ(G)}$ 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.

No reviews yet.

No comments yet.