On the Balance Index Sets of Bi-regular and Tri-regular Graphs

The degree set \( \mathcal{D}(G) \) of a graph \( G \) is the set of degrees of its vertices. It has been shown that when the cardinality of \( \mathcal{D}(G) \) is \( 1 \) (i.e., \( G \) is regular) or \( 2 \) (i.e., \( G \) is bi-regular), the balance index set of \( G \) has simple structures. In this work, we determine the balance index sets of unicyclic graphs and subclasses of \( (p, p+1) \) graphs to demonstrate the application of this recent result. In addition, we give an explicit formula for the balance index sets of subclasses of complete tri-bipartite graphs \( G \) (\(|\mathcal{D}(G)| = 3\)). Structural properties regarding the balance index sets of a general graph \( G \) and application examples are also presented.