On Energy of Prime Ideal Graph of a Commutative Ring Associated with Transmission-Based Matrices

This research explores the energy of the prime ideal graph of a commutative ring. The study demonstrates the energy formula of the graph associated with transmission-based matrices. Through research, the findings highlight the distance, Wiener-Hosoya, and distance signless Laplacian matrices. It should be noted that the distance and Wiener-Hosoya energies are always twice their spectral radius, meanwhile, it does not hold for distance signless Laplacian energy.