Characterization of certain families of modular multiplicative divisor graphs

A modular multiplicative divisor (MMD) labelling of a graph G with p vertices and q edges is a bijection f from the set of all vertices to the set of positive integers {1, 2, …, p} such that the sum of all edge labels f*(uv) = f(u)f(v)(modp); uv ∈ E(G) is a multiple of p. In this paper, we characterize certain classes of MMD graphs. This result is a solution to the open problem posed by G. Sethuraman in AKCE International Journal of Graphs and Combinatorics, No. 1 (2009) 229-236. In addition, we discuss the role of MMD labelling in obtaining cyclic decompositions of certain families of graphs into specific graphs.