This thesis has three objectives. The first is to complete what Fayres, Mahmood, and Mohammed each independently did in adding a column and then columns to the e-abacus diagram, so that we could add a row and then rows on the same diagram this time and know the requirements for it. The second objective is to locate a new movement on the diagram that has never been used before, to characterize it, and to determine what future possibilities it presents. The ultimate and most significant goal, which was to find an industrial application that could profit from them, was served by both of the aforementioned objectives. We achieved this through the work that Kirani presented in his research on a textile application in 2022, which was based on a model submitted for a doctoral dissertation in the United States of America by Mukai in 2019.

In the past years, there was an attempt to add one column to the e-abacus diagram, then another work followed by other researchers by adding several columns to the same diagram. At that time, a question remained, is it possible to add a row and later several rows on the same diagram? Or is this not an easy thing except under certain conditions? In this research, we will know when we can take these steps.

We will provide a new method in this study that integrates two types of applications, namely Graph Theory and Conjugate Young Diagram, the idea of combining the graph and the Young diagram is presented by Ali And Mahmood, which is primarily based on the idea of the e-abacus diagram, the new method is called GCYD, it directly applies to the English letter section, which will be a two-layer coding. It makes it difficult to detect the word or sentence.

The question often revolves around the extent to which the fields of mathematics can be applied, and is it possible to benefit from them in our lives? We try to provide direct answers to some of them, while we are unable to provide answers to other areas. However, in this work we will present a new vision different from everything that was presented on the subject of the e-abacus diagram; by using a vertical reading which is not previously used for any partition instead of the horizontal, where the works of each Hann and Thomas, Mukai and recently Kirani; each separately, in finding mathematical relationships with cotton and industrial products inspired us to find a link more than Wonderful, it will be useful in increasing production or the possibility of presenting one or more models for design through this diagram.