A Study of Mathematics Graph Theory with Path Matrices Theorem

Exploring the Matrix Properties and Theorems of Graph Theory

Abstract

This paper explores the theory of graphs, their associated images and the matrix properties found in modern algebra. The most interesting examples found in incidence matrices, trajectory matrices, distance matrices and the Laplacian matrices are discussed as well as the adjacent graph matrices. Work involves the use of matrix representatives, including decoupled graphics, complete graphs and trees for various graph groups. This article discusses some of the most important theorems to accomplish this aim in matrix representations of graphs. The graphs are an incredibly flexible device, since they can all model the complexities of linguistic relations and universality of modern algebraic structures from modern computer science and geographic complexity. The depiction of these graphs as matrices only improves the machine aspects. Finally, it is necessary to use modern algebra.