Abstract

We'll start the day by studying functions, specifically the kinds of functions that can be described by expressions with polynomials. In data modelling, the most common types of functions are polynomial functions and rational functions. With a little more research, functions like this can be broken down into subcategories. There are many ways to use these functions, such as in mathematical models of production costs, consumer demands, wildlife management, biological processes, and a wide range of other scientific investigations and research projects. These operations are also used a lot in many different other situations. By using these algorithms and the graphs they create, you can figure out what the data is likely to do in the future. Because it can be done, it shows that it is possible. We look at graph polynomials, graph transformations, and distance-related ideas in the context of algebraic graph theory. With new software that is still being made, it will be possible to figure out the distance polynomials of graphs with up to 200 vertices. The algorithm also finds out what the distance matrix's eigenvalues and eigenvectors are. With just the information about the neighbourhood, the method can make a "distance matrix." The Givens-Householder method is used to figure out the eigenvalues and eigenvectors, and the author's own programmes are used to figure out the characteristic polynomials of the distance matrix. New programmes are tested on a large number of graphs with a lot of vertices to make sure they work as planned. Even though distance polynomials are not usually unique structural invariants, it has been shown that they can be used to tell the difference between certain classes of cyclic isospectral graphs.

Description