2020/05/18 by Samanta, Aniruddha, Kannan, M. Rajesh · 1 citation
#05C22 #05C35 #05C50 #Combinatorics (math.CO) #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2005.08634
A \mathbbT-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 E(Φ) of a \mathbbT -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 \mathbbT -gain graph, and establish bounds for it. For any \mathbbT -gain graph Φ, we prove that 2τ(G)-2c(G) ≤ E(Φ) ≤ 2τ(G)√(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 \mathbbT -gain graphs for which E(Φ)=2τ(G)-2c(G) holds. Also, we characterize the classes of \mathbbT -gain graphs for which E(Φ)= 2τ(G)√(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.