Matching some graph dimensions with special generating functions

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.