Relations among the graph,knot graph and matrices of graph and applications

Graph theory is used in many fields such as data analysis, computer science, electrical and electronics. It is known that graphs are mapped to matrices with different methods such as neighborhood, directed neighborhood, degree, Laplacian for algebraic representation. It is also known that graphs are mapped to knots using the Tait method, which has many applications in many fields. This study aims to combine these two factors. In other words, firstly, it is aimed to obtain the graph corresponding to a knot and the matrices corresponding to this graph, and conversely, to obtain the graph corresponding to a matrix and the knot corresponding to this graph. In the first two sections of this paper, literature information is given. In the third section, basic concepts and theorems about graphs and knots are introduced. In the fourth section, the methods of obtaining matrices from graphs are discussed. In the fifth section, it is shown how to obtain a matrix from a knot with the help of a knot graph and conversely how to obtain a knot from a matrix with the help of the corresponding graph. In addition, simple applications of problems that we may encounter in our daily lives are included in this chapter. In the sixth and last section, it is concluded that there is no correlation between the coloring of graphs and the coloring of knots. Moreover, the neighborhood and directional neighborhood matrices corresponding to (2,n) tor knots are formulated.