Fuzzy Matrices with Uncertain Rows and Columns

In FMs, it is seen that the size of the matrix is fixed, and it is presumed that all columns and rows are certain; only the elements are uncertain. But, in many real-life applications, it is seen that columns and rows are not necessarily certain. For example, a fuzzy graph can be represented as a fuzzy matrix called an adjacent matrix. In the adjacent matrix for each vertex, there is a column and a row, e.g. for the sth vertex, there is sth row and sth column in the FM. All the rows and columns of the adjacent matrix are certain for a crisp graph, as the vertices and edges of a crisp graph are certain. On the other hand, the vertices and edges are uncertain in a fuzzy graph. Thus, the rows and columns in the adjacent matrix corresponding to the vertices are uncertain. In this case, the membership values of vertices are assigned as the membership values of the columns and rows. This motivates defining a new FM type with uncertain columns and rows. In 2016, Pal defined this new kind of matrix and referred to this FM as fuzzy matrices with uncertain columns and rows (FMURCs). This is a completely new idea in the FM theory.