Graph theory is a branch of discrete mathematics that is used to model and analyze interconnected systems. This paper presents a review on application of its concepts and algorithms that support diverse applications in computer science, biology, neuroscience, social sciences, engineering and geosciences. It is also reviewedthat graph theory helped in network optimization, clustering, molecular modelling and brain connectivity, it provides powerful tools for solving complex problems. It is reviewed that spectral methods, probabilistic models and graph neural network let the graph theory to evolve continuously as an essential interdisciplinary framework for understanding and optimizing complex networks in the modern world.