This Paper Explores Graph Theory, Their Associated Matrix Representations, and the Matrix Properties Found In Modern Algebra. Not Only the Adjacent Graph Matrices But Also the Most Interesting Examples Found In Incidence Matrices, Trajectory Matrices, the Distance Matrices and the Laplacian Matrices Are Discussed. Work Includes the Use of Matrix Representations For Various Graph Groups, Including Decoupled Graphs, Complete Graphs and Trees. This Paper Discusses some of the Most Important Theorems In Matrix Representations of Graphs to Accomplish This Objective. the Graphs Are an Incredibly Flexible Device, Since They Can All Models from Modern Informatics and Geographical Complexity to the Complexities of Language Relations and the Universality of Modern Algebraic Structures. the Representation of These Graphs As Matrices Enhances the Machine Aspects of This Modeling Only. In the End, Modern Algebra Is Needed.