Balance Index Set of Caterpillar and Lobster Graphs
Pradeep G. Bhat, Devadas C Nayak · 2016
Let G be a graph with vertex set V (G) and edge set E(G). Consider the set A = {0, 1}. A labeling f : V (G) → A induces a partial edge labeling f∗ : E(G) → A defined by f∗(xy) = f(x), if and only if f(x) = f(y) for each edge xy ∈ E(G). For i ∈ A, let vf (i) = |{v ∈ V (G) : f(v) = i}| and ef∗(i) = |{e ∈ E(G) : f∗(e) = i}|. A labeling f of a graph G is said to be friendly if |vf (0) − vf (1)| ≤ 1. A friendly labeling is balanced if |ef∗ (0)−ef∗(1)| ≤ 1. The balance index set of the graph G,BI(G), is defined as {|ef∗ (0) − ef∗(1)| : the vertex labeling f is friendly}. In this paper, we obtain the balance index set of caterpillar graphs and lobster graphs.