On The Balance Index Sets of Graphs (original) (raw)

Let G be a graph with vertex set V(G) and edge set E(G), and let A ={O, I}. A labeling f: V(G) -7 A induces a partial edge labeling f* : E(G) -7 A defined by f*(xy) = f{x), if and only iff{x) f{y), for each edge xy E E(G). For i E A, let vt<i) = card{v E V(G): f{v) i} and er-(i) card{e E E(G): f"(e) = i}. A labeling f of a graph G is said to be friendly if I vt<0) vt< I) I=: ;; 1. If, let< 0) et<l) I =:;; I then G is said to be balanced. The balance index set of the graph G, Bl(G), is defined as {let<O) -et<1)1 : the vertex labeling fis friendly}. Results parallel to the concept offriendly index sets are presented.