International Journal of Computer Applications (0975 - 8887) Volume 52 - No. 13, August 2012 Balanced Labeling and Balance Index Set of One Point Union of Two Complete Graphs Pradeep G Bhat Department of Mathematics Manipal Institute of Technology Manipal, Karnataka, India Devadas Nayak C Department of Mathematics Manipal Institute of Technology Manipal, Karnataka, India ABSTRACT Let G be a graph with vertex set V (G) and edge set E(G), and 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 v f (i)= |{v ∈ V (G): f (v)= i}| and e f ∗ (i)= |e ∈ E(G): f ∗ (e)= i|. A labeling f of a graph G is said to be friendly if |v f (0) − v f (1)|≤ 1. A friendly labeling is called balanced if |e f ∗ (0) − e f ∗ (1)|≤ 1. The balance index set of the graph G, Bl(G), is defined as {|e f ∗ (0) − e f ∗ (1)|: the vertex labeling f is friendly}. We provide balanced labeling and balance index set of one point union of two complete graphs. General Terms: Balance index set of graph G is denoted by BI(G) and one point union of two complete graphs is denoted by K m · K n . Keywords: Vertex labeling, Friendly labeling, Cordial labeling, Balanced la- beling and Balance index set. 1. INTRODUCTION We begin with simple, finite, connected and undirected graph G=(V,E). Here elements of set V and E are known as vertices and edges respectively. If G and H are graphs with the property that the identification of any vertex of graph G with an arbitrary vertex of graph H results in a unique graph(up to isomorphism), then we write G · H for this graph. This graph is known as one point union of two graphs. For all other terminologies and notations we follow Harary [?]. If the vertices of the graph are assigned values subject to certain conditions is known as graph labeling. Most interesting graph labeling problems have three important characteristics. ∗ a set of numbers from which the labels are chosen; ∗ a rule that assigns a value to each edge; ∗ a condition that these values must satisfy. For detail survey on graph labeling one can refer Galian [?]. Vast amount of literature is available on different types of graph labeling. According to Beineke and Hegde [?] graph la- beling serves as a frontier between number theory and structure of graphs. Labeled graphs are becoming an increasingly useful fam- ily of mathematical models for a broad range of applications. The qualitative labelings of graph elements have inspired research in diverse fields of human enquiry such as conflict resolution in social psychology, electrical circuit theory and energy crisis. Quantitative labelings of graphs have led to quite intricate fields of applications such as coding theory problems, including the design of good radar location codes, synch-set codes, missile guidance codes and convolution codes with opti- mal auto-correlation properties. Labeled graphs have also been applied in determining ambiguities in X-Ray crystallographic analysis, to design communication network addressing systems, to determine optimal circuit layouts and radio-astronomy, etc. In 1986 Cahit [?] introduced cordial graph labeling. A function f from V (G) to {0, 1} , where for each edge xy, f ∗ (xy) = |f (x) − f (y)|, v f (i) is the number of vertices v with f (v)= i, and e f ∗ (i) is the number of edges e with f ∗ (e)= i is called friendly if |v f (0) − v f (1)|≤ 1. A friendly labeling f is called cordial if |e f ∗ (0) − e f ∗ (1)|≤ 1. Lee, Liu and Tan [?] considered a new labeling problem of graph theory. A vertex labeling of G is a mapping f from V (G) into the set {0, 1} . For each vertex labeling f of G,a partial edge labeling f ∗ of G is defined in the following way. For each edge uv in G, f ∗ (uv)=  0, if f (u)= f (v)=0 1, if f (u)= f (v)=1 Note that if f (u) = f (v), then the edge uv is not labeled by f ∗ . Thus f ∗ is a partial function from E(G) into the set {0, 1}. Let v f (0) and v f (1) denote the number of vertices of G that are labeled by 0 and 1 under the mapping f respectively. Likewise, let e f ∗ (0) and e f ∗ (1) denote the number of edges of G that are labeled by 0 and 1 under the induced partial function f ∗ respectively. DEFINITION 1. A graph G is said to be a balanced graph or balanced if there is a vertex labeling f of G such that |v f (0) − v f (1)|≤ 1 and |v f ∗ (0) − v f ∗ (1)|≤ 1. A graph G is said to be strongly vertex-balanced if G is a balanced graph and v f (0) = v f (1). Similarly a graph G is strongly edge-balanced if it is a balanced graph and e f ∗ (0) = e f ∗ (1). If G is a strongly vertex-balanced and strongly edge-balanced graph, then G is a strongly balanced graph. EXAMPLE 1. Figure 1 shows a graph with two distinct balanced labelings of graph G. The following graphs are studied in [?] (1) The path P n is balanced; it is strongly balanced if n is even. (2) The cycle C n is balanced; it is strongly balanced if n is even. 1