Diophantine Edge Graceful Graph

Background: Graph labeling problems have interesting applications in coding theory, communication networks, optimal circuits’ layouts and graph decomposition problems, as described in various patents. Consider a graph G = (V,E), with |V| = p and |E| = q.If f:V → {0,1,2,.....p} is a bijective mapping and if f+ : E → {1,2,... .q} be defined by f+ (uv) = |f(u) - f(v)|, for u, v ε V, and if f+ is bijective, then the induced map f+ gives an edge graceful labeling. Methods: In this paper, we label edge labels directly, taking labels from the solutions of a relevant Diophantine equations. This edge labeling has two steps: first we label vertices by f, then we induce labeling to edges. We have considered patents “System and method for making decisions using network-guided decision trees with multivariate splits” and “Graph-based ranking algorithms for text processing’” Results: In section 1, we have some results on complete (m, h) trees, (m ≥ 2, h ≥ 1 ) leading to graceful, odd-edge graceful and almost edge-graceful labeling. In section 2, we have edge-labeled corona graphs. Conclusion: A compact representation for the graph given in the form of integers which may be used in the application of graphs for which Diophantine Edge Graceful labeling is possible. The possibility of Diophantine Edge Graceful labeling of a graph may be known by structural properties of graphs .