Geometric Decomposition of Graphs with Various Common Ratios

Abstract

Let G = (V,E) be a simple connected graph with p vertices and q edges. A decomposition of G is a collection of edge disjoint subgraphs G1,G2,...,Gn of G such that every edge of G belongs to exactly one Gi. A decomposition Ga,Gar,Gar2,Gar3,...,Garn−1 of G is said to be a Geometric Decomposition(GD) if each Gari−1 is connected and |E (Gari−1)| = ari−1, for every i = 1,2,3,...n and a,r ∈ N.Clearly q = a(rn−1) r−1 . In this paper we give Geometric Decomposition of graphs with first term as 1 and 3 ie., a = 1,3 along with various common ratio.The Geometric Decomposition of graphs such as Fan graph, K1,m + K1, Ladder graph, Book graph, Bistar graph, Friendship graph are found