Balance Index Set of Caterpillar and Lobster Graphs

We begin with simple, finite, connected and undirected graph G=(V,E). Here the elements of set V and E are known as vertices and edges respectively with |V | = p and |E| = q. For all other terminologies and notations we follow Harary [1]

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) ifand 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