Bounds and extremal graphs for the energy of complex unit gain graphs

Article Type

Research Article

Publication Title

Linear Algebra and Its Applications

Abstract

A complex unit gain graph (T-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 T-gain graph Φ is the sum of the absolute values of all eigenvalues of A(Φ). For any connected triangle-free T-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)≤2μ2Δe+1 and characterize the classes of graphs for which the bound is 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 T-gain graphs defined on G.

First Page

844

Last Page

866

DOI

10.1016/j.laa.2024.03.028

Publication Date

9-15-2025

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