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 ∗−1(1)| ∣∣ |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.