Balanced Labeling and Balance Index Set of One Point Union of Two Complete Graphs

Abstract

Let G be a graph with vertex set V (G) and edge set E(G), and consider the set A = f0;1g. 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 xy2 E(G). For i2 A, let vf(i) =jfv2 V (G) : f(v) = igj and ef (i) =je2 E(G) : f (e) = ij. A labeling f of a graph G is said to be friendly if jvf(0) vf(1)j 1. A friendly labeling is called balanced ifjef (0) ef (1)j 1. The balance index set of the graph G, Bl(G), is defined asfjef (0) ef (1)j: the vertex labeling f is friendlyg. We provide balanced labeling and balance index set of one point union of two complete graphs.