2023/12/28 by Aniruddha Samanta, M. Rajesh Kannan, Samanta, Aniruddha +1
Mathematics · Computer Science · Materials Science · #Graph theory and applications #Matrix Theory and Algorithms #Graphene research and applications
paper · pdf · doi:10.48550/arxiv.2312.17152
A complex unit gain graph ( \mathbbT -gain graph), Φ=(G, φ) is a graph where the gain function φ 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(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) . For any connected triangle-free \mathbbT -gain graph Φ with the minimum vertex degree δ, we establish a lower bound E(Φ)≥ 2δ and characterize the equality. Then, we present a relationship between the characteristic and the matching polynomial of Φ. Using this, we obtain an upper bound for the energy E(Φ)≤ 2μ√(2Δe+1) and characterize the classes of graphs for which the bound sharp, where μ and Δe are the matching number and the maximum edge degree of Φ, respectively. Further, for any unicyclic graph G , we study the gains for which the gain energy E(Φ) attains the maximum/minimum among all \mathbbT -gain graphs defined on G.