In this work, we investigated the relationship between special generating functions, such as array polynomials and graph dimensions, including metric, multiset, outer multiset, and local multiset dimensions, using minimal monoid presentations. The present paper is founded on earlier contributions and we are concerned with resolving the matching issue. In this context, our focus is on the matching between graph dimensions and generator functions, a subject that has not been examined and is alluded to in Open problem 3 in the paper by A. S. Cevik [Matching some graph dimensions with special presentations, Montes Taurus J. Pure Appl. Math., 6 (2024), 78-89]. As part of this effort, we address the characterization of graphs with infinite multiset dimensions and provide a partial classification based on outer multiset, local multiset, and metric dimensions.

In crisp graph theory, there are numerous topological indices accessible, including the Misbalance Prodeg Index, which is one of the most well-known degree-based topological indexes. In crisp graphs, both vertices and edges have membership values of $1$ or $0$, whereas in fuzzy graphs, both vertices and edges have different memberships. This is an entire contrast to the crisp graph. In this paper, we introduce the Fuzzy Misbalance Prodeg Index of a fuzzy graph, which is a generalized form of the Misbalance Prodeg Index of a graph. We find bounds of this index and find bounds of certain classes of graphs such as path graph, cycle graph, complete graph, complete bipartite graph, and star graph. We give an algorithm to find the Fuzzy Misbalance Prodeg Index of a graph for the model of multi-criteria decision-making is established. We present applications from daily life in multi-criteria decision-making. We apply our obtained model of the Fuzzy Misbalance Prodeg Index for the multi-criteria decision-making to the choice of the best supplier and we also show the graphical analysis of our index with the other indices that show how our index is better than other existing indices.