The Harmonic Graphs Conjecture states that there exists an asymptotic relation involving the Harmonic Index and the natural logarithm as the order of the graph increases. This conjecture, grounded in the novel context of Prime Graphs, draws upon the Prime Number Theorem and the sum of divisors function to unveil a compelling asymptotic connection. By carefully expanding the definitions of the harmonic index and the sum of divisors function, and leveraging the prime number theorem's approximations, we establish a formula that captures this intricate relationship. This work is an effort to contribute to the advancement of graph theory, introducing a fresh lens through which graph connectivity can be explored. The synthesis of prime numbers and graph properties not only deepens our understanding of structural complexity but also paves the way for innovative research directions.

Abstract: Thermal desalination is yet a reliable technology in the treatment of brackish water and seawater; however, its demanding high energy requirements have lagged it compared to other non-thermal technologies such as reverse osmosis. This review provides an outline of the development and trends of the three most commercially used thermal or phase change technologies worldwide: Multi Effect Distillation (MED), Multi Stage Flash (MSF), and Vapor Compression Distillation (VCD). First, state of water stress suffered by regions with little fresh water availability and existing desalination technologies that could become an alternative solution are shown. The most recent studies published for each commercial thermal technology are presented, focusing on optimizing the desalination process, improving efficiencies, and reducing energy demands. Then, an overview of the use of renewable energy and its potential for integration into both commercial and non-commercial desalination systems is shown. Finally, research trends and their orientation towards hybridization of technologies and use of renewable energies as a relevant alternative to the current problems of brackish water desalination are discussed. This reflective and updated review will help researchers to have a detailed state of the art of the subject and to have a starting point for their research, since current advances and trends on thermal desalination are shown.