Distance spectrum and energy in single valued neutrosophic graphs

This study introduces the concept of distance matrix and distance energy in the context of single valued neutrosophic graphs (SVNGs), which extend classical graph theory by incorporating uncertainty and indeterminacy through neutrosophic sets. In SVNGs, edge weights are defined by truth, indeterminacy, and falsity membership degrees, while the distance matrix captures the shortest path lengths between vertex pairs. The distance energy, derived from the eigenvalues of this matrix, reflects the structural properties of the graph under uncertainty. We establish theoretical upper and lower bounds for the distance energy of SVNGs, offering insight into its behavior. To demonstrate practical utility, we apply this framework to the traveling salesman problem, modeling it with an SVNG to account for uncertain travel costs. By computing and comparing the distance energy of various paths, we rank them using a method that emphasizes truth while incorporating indeterminacy and falsity. The results highlight the applicability of distance energy in uncertain environments and underscore its value as a decision-support tool in optimization problems involving imprecise or incomplete data.