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The paper establishes some new bounds for Randic and GA indices.

A digraph D with p vertices and q arcs is labeled by assigning a distinct integer value g(v) from f0, 1, :::, qg to each vertex v. The vertex values, in turn, induce a value g(u, v) on each arc (u, v) where g(u, v) ¼ (g(v) À g(u)) (mod q þ 1) If the arc values are all distinct then the labeling is called a graceful labeling of digraph. In this survey article, we have collected results that we could find interesting on graceful labeling of digraphs.

A directional labeling of an edge uv in a graph G = (V, E) by an ordered pair ab is a labeling of the edge uv such that the label on uv in the direction from u to v is ℓ(uv) = ab, and ℓ(vu) = ba. New characterizations of 2-colorable (bipartite) and 3-colorable graphs are obtained in terms of directional labeling of edges of a graph by ordered pairs ab and ba. In addition we obtain characterizations of 2-colorable and 3-colorable graphs in terms of matrices called directional adjacency matrices.