Energy Bounds of Soft Graphs

Abstract

The paper examines energy in soft graphs and establishes the maximum energy such graphs can contain. It introduces several key concepts, namely equi-energy soft graphs, hyper-energy soft graphs, non-hyper-energy soft graphs, and hypo-energy soft graphs. These amounts of energy less than the number of their vertices are called hypo-energy soft graphs, and their counterparts, which surpass complete graph energy, are called hyper-energy; these categories display structural properties under uncertainty, and find use in the analysis of networks. These concepts facilitate a deeper understanding of energy within the framework of soft graph theory. The paper also investigates the maximum and minimum energy levels achievable by soft graphs, taking into account variables such as the number of vertices and edges. Furthermore, it tackles the challenges associated with computing eigenvalues for the adjacency matrices of soft graphs. The paper substantiates these concepts with diverse examples that demonstrate their application and importance. These findings extend traditional graph energy theory to address uncertain scenarios and open new avenues for research in soft computing and network analysis.