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

摘要

The degree set V(G) of a graph G is the set of degrees of its vertices. It has been shown that when the cardinality of V(G) is 1 (i.e. G regular) or 2 (i.e. G 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 (|D(G)| = 3). Structural properties regarding the balance index sets of a general graph G and application examples are also presented.