BALANCED INDEX SETS OF GRAPHS AND SEMIGRAPHS

Let G be a simple graph with vertex set V (G) and edge set E(G). Graph labeling is an assignment of integers to the vertices or the edges, or both, subject to certain conditions. For a graph G(V, E), a friendly labeling f : V (G) → {0, 1} is a binary mapping such that |v_f(1) − v_f(0)| ≤ 1, where v_f(1) and v_f(0) represents number of vertices labeled by 1 and 0 respectively. A partial edge labeling f^∗ of G is a labeling of edges such that, an edge uv ∈ E(G) is, f^∗(uv) = 0 if f(u) = f(v) = 0; f^∗(uv) = 1 if f(u) = f(v) = 1 and if f(u) ≠ f(v) then uv is not labeled by f^∗. A graph G is said to be balanced graph if it admits a vertex labeling f that satisfies the conditions, |v_f(1) − v_f(0)| ≤ 1 and |e_f(1) − e_f(0)| ≤ 1, where e_f(0), e_f(1) are the number of edges labeled with 0 and 1 respectively. The balanced index set of the graph G is defined as, {|e_f(1) − e_f(0)| : the vertex labeling f is friendly}. A semigraph is a generalization of graph. The concept of semigraph was introduced by E. Sampath Kumar. Frank Harrary has defined an edge as a 2-tuple (a, b) of vertices of a graph satisfying, two edges (a, b) and (a^′, b^′) are equal if and only if either a = a^′ and b = b^′ or a = b^′ and b = a^′. Using this notion, E. Sampath Kumar defined semigraph as a pair (V, X) where V is a non-empty set whose elements are called vertices of G and X is a set of n-tuples called edges of G of distinct vertices, for various n ≥ 2 satisfying the conditions: (i) Any two edges of G can have at most one vertex in common; and (ii) two edges (a1, a2, a3, ..., ap) and (b1, b2, b3, ..., bq) are said to be equal if and only if the number of vertices in both edges must be equal, i.e p = q, and either ai = bi for 1 ≤ i ≤ p or ai = bp−i+1, 1 ≤ i ≤ p. In this article, balance index set of T(Pn), T(Wn), T(Km,n) and T(Sn) determined, and the balance index set of semigraph is introduced. Additionally, the balanced index set of semigraph Cn,m, Kn,m is determined.