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

Abstract

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.