ON THE GRAPHS D(Γ(M )) AND D(Λ(M )) OF MODULES

Abstract

In this paper, we introduce two new graphs to unital modules M over commutative rings R called the double to- tal graph with respect to singular submodule which is de- noted by D(Γ(M )) and the double total graph of M over R with respect to the set of torsion elements of M which is denoted by D(λ(M )). We find the connectedness, di- ameter, girth, independence number, domination number of D(Γ(M )). We determine when is the graph D(Γ(M )) a Hamiltonian graph, a regular graph and also, we find the clique number of this graph. When M is a torsionable mod- ule, we investigate when is the graph D(λ(M )) complete bipartite.