An algebraic approach for finding balance index sets

Any vertex labeling f : V →{0,1} of the graph G = (V,E) induces a partial edge labeling f* : E → {0,1} defined by f*(uv)= f (u)if and only if f (u)= f (v). The balance index set of G is defined as {\f _*-1 (0) - f| \f_-1 (0) - f_-1 (1) ≤1}. In this paper, we propose a new and easier approach to find the balance index set of a graph. This new method makes it possible to determine the balance index sets of a large number of families of graphs in an unified and uniform manner.