Signed Quantum Graphs: A Dynamical System

Traditional Graphs, such as weighted graphs, or fuzzy graphs have static edge weights or membership values. These graphs have been widely used to model various complex systems, but they typically do not incorporate dynamic interactions found in evolving systems like brain networks. Temporal graphs are data-based algorithmic time-dependent structures. There is no such structure to represent the dynamic interactions of networks. Quantum graphs are structured graphs based on the dynamic behaviors of vertices and links. This study introduces Signed Quantum Graphs (SQGs) with time-dependent signs that represent dynamical systems. The properties of SQGs have been investigated. Additionally, concepts and properties of triadic closure, balance, domination, and strength of a SQG have been studied. The area of applications in brain networks has been mentioned. We propose a dynamic signed graph framework, where edge signs evolve periodically using functions. Structural properties and convergence results are provided.