Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has caused the global pandemic, coronavirus disease-2019 (COVID-19) which has resulted in 60.4 million infections and 1.42 million deaths worldwide. Mathematical models as an integral part of artificial intelligence are designed for contact tracing, genetic network analysis for uncovering the biological evolution of the virus, understanding the underlying mechanisms of the observed disease dynamics, evaluating mitigation strategies, and predicting the COVID-19 pandemic dynamics. This paper describes mathematical techniques to exploit and understand the progression of the pandemic through a topological characterization of underlying graphs. We have obtained several topological indices for various graphs of biological interest such as pandemic trees, Cayley trees, Christmas trees, and the corona product of Christmas trees and paths. We have also obtained an analytical expression for the thermodynamic entropies of pandemic trees as a function of R0, the reproduction number, and the level of spread, using the nested wreath product groups. Our plots of entropy and logarithms of topological indices of pandemic trees accentuate the underlying severity of COVID-19 over the 1918 Spanish flu pandemic. © 2020 by the authors. Licensee MDPI, Basel, Switzerland.